title: On General Extension Fields for the Classical Groups in Differential Galois Theory
authors: Matthias Seiss
text: 
  
  
  Let G be one of the classical groups of Lie rank l. We make a similar 
  construction of a general extension field in differential Galois theory 
  for G as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential 
  field E of differential transcendence degree l over the constants 
  on which the group G acts and show that it is a Picard-Vessiot extension of the field of invariants E^G.
  The field E^G is differentially generated by l differential 
  polynomials which are differentially algebraically independent over the constants. They are the coefficients of the defining equation of the extension. 
  Finally we show that our construction satisfies generic properties for a specific kind of 
  G-primitive Picard-Vessiot extensions.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  : Semantic API Migrations
      Michael F.P. O'Boyle
      December 30, 2023
  =========================
  
  
  
  
  
  
  § INTRODUCTION
  
  
  In classical Galois theory there is a well-known construction of the general 
  equation with Galois group the symmetric group S_n. As a starting point 
  one takes n indeterminates T=(T_1,…,T_n) 
  and considers the rational function field ℚ(T).
  The group S_n acts on 
  ℚ(T) by permuting the indeterminates and one can show  
  that ℚ(T) is a Galois extension of the field of invariants 
  ℚ(T)^S_n for a polynomial equation of degree n whose coefficients 
  are the elementary symmetric polynomials. The field of invariants is 
  generated by these polynomials and they are algebraically independent over ℚ. 
  A generalization of this idea leads to 
  the so called Noether problem
  which would solve the inverse problem over ℚ for finite groups.
  Unfortunately, the answer to the Noether problem is not affirmative.
  
  
  In differential Galois theory one can construct in a similar way a general equation with 
  differential Galois group the full general linear group GL_n(C). 
  For an algebraically closed field C of characteristic zero and n differential 
  indeterminates y=(y_1,…,y_n) one begins with the 
  differential field C⟨y⟩, that is the field 
  differentially generated by the indeterminates y over C. 
  The action of the group GL_n(C) 
  on C⟨y⟩ is induced by linear transformations 
  from the right on the wronskian matrix in y.
  As in classical Galois theory one can show that C⟨y⟩ 
  is a Picard-Vessiot extension of the field of invariants 
  C⟨y⟩^GL_n(C) 
  with differential Galois group GL_n(C).
  As in the case of the symmetric group this extension is defined by a linear 
  differential equation whose coefficients are differentially algebraically independent over C
  and differentially generate the field of invariants.
  This idea was generalized in <cit.> by L. Goldman and he applied the techniques 
  developed there to some connected subgroups 
  G ⊂GL_n(C) obtaining explicit equations. 
  
  In the present work we perform a similar construction in differential Galois theory 
  for the classical groups. For this purpose let G be one of these groups and denote by 
  l its Lie rank.
  As in the above example we start our construction 
  with a differential field C⟨η⟩ which is differentially 
  generated by l differential 
  indeterminates η=(η_1,…,η_l) over the constants C. We  
  use this purely differential transcendental extension to build our final general extension field 
  E ⊃ C⟨η⟩ by taking into 
  account the structure of G-primitive Picard-Vessiot extensions.  
  These are Picard-Vessiot extensions with  
  differential Galois group G whose fundamental solution matrices satisfy the algebraic relations defining 
  the group G.
  The structural information of these extensions is obtained by connecting results from the theory 
  of reductive groups with differential Galois theory.
  More precisely, the Bruhat decomposition provides a normal form of elements of G 
  parameterized by a fixed Borel subgroup B and the Weyl group. It turns out that 
  the Bruhat decomposition of a fundamental matrix Y̅ of a G-primitive Picard-Vessiot extension
  is with respect to the maximal Weyl group element w̅, stemming from the fact that Bw̅B 
  is a rational variety. Using the normal form Y̅=u_1 w̅ t u_2, where u_1, u_2 are 
  elements of the maximal unipotend subgroup of B and t lies in the torus, we found out that the entries of 
  t and u_2 generate a Liouvillian extension over the fixed field under the Borel subgroup, which on his 
  part is generated by the entries of u_1.
  In a specific case when the defining 
  matrix has shape as constructed in <cit.>, the fixed field is 
  differentially generated by Lie rank many elements x=(x_1,…,x_l) and the Liouvillian extension 
  can be parameterized by these 
  elements, that is, it is generated by l exponentials, whose logarithmic derivatives are x, and 
  successive integrals of these exponentials. As a consequence the whole Picard-Vessiot extension is determined by the l 
  elements x. According to these observations we construct our general extension field E. We use the indeterminates η to parameterize a general element of the 
  Cartan subalgebra and this element together with the basis elements of root spaces belonging to the negative simple roots 
  define the Liouvillian extension E of C⟨η⟩ with 
  fundamental matrix t u_2 which lies in the Borel group.
  We show that we can build an element u_1 in the maximal unipotent subgroup with entries in 
  C⟨η⟩ such that the logarithmic derivative of 
  Y=u_1 w̅ t u_2 has shape as the matrix equations constructed in <cit.>.  
  Similar as in the case of GL_n(C) the group G acts on E by right multiplication on Y, but now we take the Bruhat decomposition to obtain the effect on the generators of E. 
  We prove that E is a Picard-Vessiot extension of the 
  field of invariants E^G with differential Galois group G and that  
  E^G is generated by l invariants 
  h=(h_1(η),…,h_l(η)) which 
  are differentially algebraically independent over the constants. The coefficients of the 
  linear differential equation defining our extension   
  are differential polynomials in these invariants. 
  In opposition to the above approach we started in <cit.> with a differential 
  field C⟨t⟩ generated by 
  l differential indeterminates t=(t_1,…,t_l) as our differential base field.
  Under the consideration of the geometrical structure 
  of the group G, we defined a parameter differential equation
  ∂(y)=A_G(t)y and 
  showed that it defines a Picard-Vessiot extension of C⟨t⟩ 
  with differential Galois group G. 
  Our extension field E is build in such away that its defining equation 
  is actually ∂(y)=A_G(h)y, that is the 
  indeterminates t can be taken to be the differentially algebraically 
  independent invariants h. In fact we have shown that for the 
  parameterized equations of <cit.> we can perform a similar construction as
  E. Noether did in classical Galois theory. 
  
  In contrast to the above example of the general equation 
  for GL_n(C), where the extension field is generated by n differential 
  indeterminates over the constants, the differential transcendence degree of the extension 
  field of our construction is equal to the Lie rank. 
  In the first case the extension and equation are clearly generic. We can substitute any n
  linearly independent solutions of any scalar linear differential equation of degree n 
  with differential Galois group a subgroup of GL_n(C) into y and 
  the general equation specializes to it. At a first view
  our construction is only generic for G-primitive Picard-Vessiot 
  extensions of a differential field F stemming from differential equations of shape 
  ∂(y)=A_G(f)y with f∈ F^l
  in the sense that we can substitute the l elements x determining the extension 
  into η which has the effect that A_G(h) specializes to 
  A_G(f). In order to be generic for a broader range of Picard-Vessiot extension 
  we consider extensions whose defining matrix is gauge equivalent to our equation. 
  In <cit.> we have seen that every matrix of an open subset of the direct sum of the Lie algebra of B^- and the root spaces corresponding to the positive simple roots 
  is gauge equivalent to a matrix of shape A_G(f). For such a defining matrix 
  the logarithmic derivative 
  of the Liouvillian part u_2 has the shape of a principal nilpotent 
  matrix in normal form (for a definition see 
  Chapter <ref>). 
  It turns out that an arbitrary defining matrix is gauge equivalent to a 
  matrix of shape A_G(f) if and only if the defining matrix of 
  its Liouvillian part  is gauge equivalent by specific 
  transformations to principal nilpotent 
  matrix in normal form. We will call such  extensions G-primitive with normalisable 
  unipotent part and our general extension field will be generic for such extensions.
  
  
  The paper is organized in the following way. 
  In Chapter <ref> we recapitulate 
  some basic facts about the 
  structure of the classical groups and their Lie algebras and introduce the 
  corresponding notation. Chapter <ref> shortly deals with 
  the decomposition of the gauge transformation into the adjoint action and the 
  logarithmic derivative.
  In Chapter <ref> 
  the connection between 
  the group structure and Picard-Vessiot extensions is established.   
  The normal form decomposition of a fundamental matrix of a 
  G-primitive Picard-Vessiot extension 
  and the resulting normal form coefficients which generate the extension are given in Proposition <ref>. 
  Using the 
  results we show in Proposition <ref> 
  that the normal form coefficients which belong to the torus and the second 
  unipotent group in the decomposition of the fundamental matrix generate 
  a Liouvillian extension and that these coefficients are exponentials and successive integrals respectively.
  In Chapter <ref>
  we construct our general extension field. We prove in Theorem <ref> that it
  is a Picard-Vessiot extension of the differential field generated by the 
  coefficients of the logarithmic derivative of the matrix Y from above and that the 
  differential Galois
  group of the extension is G. Reinterpreting the result in Remark <ref> we obtain that 
  E with the action of G has the above outlined properties.
  Every step in the construction 
  comes along with an example for the group SL_4. At the end of the 
  chapter we consider the case of the group G_2. Chapter <ref> deals with the generic 
  properties of our general extension field. For a full G-primitive Picard-Vessiot extension 
  we give in Theorem <ref> a criteria for gauge equivalence 
  of its defining matrix to a matrix of shape A_G(f) in terms of its Liouvillian 
  part. Our finial results are stated in Theorem <ref>. 
  
  
  
  
  § THE STRUCTURE OF THE CLASSICAL GROUPS
   
  
  Let G be one of the classical groups of Lie rank l over C. 
  Let Φ be the root system of G of the respective type and let Δ={α̅_1, … , α̅_l } be a basis of Φ. 
  We write Φ^+ for the set of positive and Φ^- for the set of negative roots and we set 
  m:= |Φ^+ | = |Φ^- |. 
  Every root α can be written uniquely as 
  
      α =   k_1 α̅_1 + … + k_l α̅_l
  
  with integer coefficients k_i where all k_i are nonpositive or nonnegative. 
  By the integer ht(α)= k_1 + … + k_l we mean the height of the root α.
  
  We denote by 𝒲 the Weyl group of Φ, that is the finite group 
  which consists of the orthogonal reflections 
  
      w_α(β) = β - ⟨β , α⟩α
  
  for α,  β∈Φ 
  where  
  ⟨β , α⟩=2(β,α)(α,α)^-1 and (·,·) denotes the inner product of the corresponding ambient real vector space of Φ. The Weyl group is generated by the reflections w_α̅_i
  for the simple roots α̅_i ∈Δ and every w_α∈𝒲
  can be written as 
  
      w_α=w_α̅_i_1⋯ w_α̅_i_k
  
  with k minimal. The integer k is called the length of w_α. 
  We denote by w̅ the unique element of maximal length, that is the element
  which maps Φ^+ to Φ^- and vice versa. 
  
  Let 𝔥 be a maximal Cartan subalgebra of the Lie algebra 
  𝔤 of G. Then with respect to 𝔥 we obtain a root space decomposition
  
      𝔤= 𝔥⊕⊕_α∈Φ𝔤_α
  
  where we denote by 𝔤_α the one-dimensional root spaces of 𝔤.   
  With respect to this root space decomposition we can choose a Chevalley basis 
  
      {H_ i|α̅_i ∈Δ}∪{ X_α|α∈Φ}
  
  where 𝔥= ⟨ H_1 , … , H_l ⟩ and 
  𝔤_α= ⟨ X_α⟩. Further we consider in the following the 
  maximal nilpotent subalgebras 𝔲= ∑_α∈Φ^+𝔤_α
  and 𝔲^-= ∑_α∈Φ^-𝔤_α
  defined by all positive and all negative roots respectively. 
  Moreover, let 𝔟^+ = 𝔥 + 𝔲^+ 
  be the maximal solvable subalgebra of 𝔤 which contains the positive 
  maximal nilpotent subalgebra 𝔲^+  
  and the Cartan subalgebra 𝔥. Analogously let 
  𝔟^- = 𝔥 + 𝔲^-. 
  
  For X,  Y ∈𝔤 we denote by [X,Y] the 
  usual bracket product and we write ad(X) for the endomorphism 
  ad(X): Y ↦ [X,Y]. For an element g ∈ G we denote by 
  
      Ad(g): 𝔤→𝔤,  X ↦ gXg^-1
  
  the adjoint action of G on 𝔤.
  
  For l nonzero elements s=(s_1,…,s_l) of C let
  
      A_0^+(s):=  ∑_i=1^l s_i X_α̅_i  and    
        A_0^-(s):=  ∑_i=1^l s_i X_-α̅_i
  
  be the sum of all basis elements which correspond to the positive simple roots 
  (resp. negative simple roots) with coefficients s. 
  We call A_0^-(s) a principal nilpotent matrix in normal form and 
  to shorten notation we write A_0^+ and A_0^- in case of s=(1,…,1). 
  
  
   We order the negative roots Φ^- = {β_1,…, β_m} such that 
  ht(β_i) ≥ht(β_i+1) for all i=1,…, m-1. 
  Then the indices of all roots of a given height form an unbroken string i_1,…,i_2. 
  Let γ_1, …, γ_l be complementary roots of Φ^- (see <cit.>). 
  For every height we reorder the roots such that the complementary roots have  the greatest indices, 
  that is, the indices of the non-complementary roots of a given height 
  form an unbroken string i_1,… i_2'. 
  
  
  We rename the basis elements 
  { X_β_i|β_i ∈Φ^- } of 𝔲^- into X_i.
  One can decompose 𝔤 into a direct sum of subspaces 
  𝔤^(j) and 𝔥 where each 𝔤^(j) is the direct sum of root spaces corresponding to roots of height j (see <cit.>). We denote
  by r(i) the negative number such that 
  X_i ∈𝔤^(r(i)), that is r(i)=ht(β_i). Then the basis is ordered such 
  that r(i) ≥ r(i+1) for all i=1,… ,m-1. For each X_i we define 
  
      W_i := [X_i, A_0^+ ].
  
  Then W_i ∈𝔤^(r(i)+1) and the set { W_i | i=1, …, m } is a basis 
  of ad(A_0^+)(𝔲^-). Further by <cit.> the set 
  
      { W_i | i=1,…,m}∪{ X_γ_k| k=1,…,l }
  
  is a basis of 𝔟^-.
  
  We denote by T the maximal torus of G whose Lie algebra is 𝔥. Further by U^+ 
  and U^- we mean the maximal unipotent groups whose Lie algebras are 𝔲^+ 
  and 𝔲^- respectively. Finally we write B^+ and B^- for the Borel 
  subgroups of G with respective Lie algebras 𝔟^+ and 𝔟^-. 
  
  For a root α∈Φ we denote by U_α the one-dimensional 
  root group of G whose Lie algebra 
  coincide with the root space 𝔤_α. There is an isomorphism 
  𝔾_a → U_α and for x ∈ C we denote by u_α(x) 
  the image of x in U_α. For the negative roots β_1,…,β_m, which 
  we ordered in a specific way,
  we rename the root groups U_β_i into U_i and their elements into u_i(x).
  For i=1,…,l let T_i be the one-dimensional 
  subtorus of T whose Lie algebra coincides with ⟨ H_i ⟩⊂𝔥. 
  Then there is an isomorphism 𝔾_m → T_i and  
  for z ∈ C^× we denote by t_i(z) the image of z in T_i.
  For m elements x=(x_1, …, x_m) of C we denote in the 
  following by u(x) the product 
  
      u(x):=u_1(x_1) ⋯ u_m(x_m)  ∈ U^-
  
  and for l elements z=(z_1, …, z_l) of C^× we write 
  t(z) for the product 
  
      t(z) :=t_1(z_1) ⋯ t_l(z_l) ∈ T.
  
   
   
  
  An important result in the structure theory of reductive groups is the Bruhat decomposition. It gives 
  a normal form for elements of G parametrized by a fixed Borel subgroup and the Weyl group.
   Fix a Borel subgroup of G. Then 
   we have G= ⋃_w ∈𝒲 B w B (disjoint union) with
   B w B = B w̃ B if and only if w = w̃ in 𝒲.
  
  
   See <cit.>.
  
  For w ∈𝒲 we fix a representative n(w) of w in the 
  normalizer N_G(T) of T in G. Following the discussion of <cit.> 
  we obtain a T-stable subgroup U'_w=U ∩ n(w)U^- n(w)^-1 of U for w. 
  A normal form for elements of G can now be made unique.
  
   For each w ∈𝒲 fix a coset representative n(w) ∈ N_G(T). 
   Then each element x ∈ G can be written in the form x=u'n(w)t u, 
   where w ∈𝒲, t ∈ T, u  ∈ U and u' ∈ U'_w
   are all determined uniquely by x. 
  
  
   See <cit.>.
  
  Borel subgroups are maximal connected solvable subgroups of G and so by 
  <cit.> 
  there is 
  a descending chain of closed subgroups of B^- where each group is a normal subgroup of its 
  predecessor and their quotient is isomorphic to 𝔾_a or 𝔾_m.  
  For i = 0, … , l-1 define T̅_i:=T_i+1⋯ T_l and denote by B̅^-_i the subgroup 
  T̅_i ⋉ U^- of B^-. Moreover for i= 0 , … ,m-1 denote by U̅^-_i the subgroup 
  U_i+1⋯ U_m and let U̅^-_m={e}. 
  Following <cit.> these groups form a descending chain of closed subgroups 
  
      B^- =B̅^-_0 ⊃B̅^-_1 ⊃…⊃B̅^-_l-1⊃U̅^-_0 ⊃U̅^-_1 ⊃…U̅^-_m-1⊃U̅^-_m={e}
  
  where each group is a normal subgroup of its predecessor and their quotient is isomorphic to 
  𝔾_a or 𝔾_m. More precisely, for i = 0, … , l-2 the quotients 
  B̅^-_i/B̅^-_i+1 and B̅^-_l-1/U̅^-_0 are isomorphic to 𝔾_m and for 
  i = 0 , … ,m-1 the quotients U̅^-_i/U̅^-_i+1 are isomorphic to 𝔾_a. 
  We denote the positive roots by Φ^+ = {α_1 , … , α_m } and we order 
  them in the same way as the negative roots. If we define as in the case of B^- and U^- 
  the subgroups B̅^+_i⊂ B^+ and U̅^+_i ⊂ U^+, then these subgroups form a 
  chain with the same properties.
  
  
  
  § THE GAUGE TRANSFORMATION
  
   
  Let F be a differential field with derivation ∂ and field of constants C. 
  In the following we mean by 
  G(F) and 𝔤(F) the group and Lie algebra of F-rational points.
  
   The map 
   
      ℓδ: GL_n(F) → F^n × n, g ↦∂(g)g^-1
  
  is called the logarithmic derivative.
  
  A gauge transformation of an element A ∈𝔤(F) by an element g of G(F)
  is defined as   
  
      Ad(g)(A) + ℓδ(g) .
  
  We can decompose the gauge transformation of A into  
  the image Ad(g)(A) of A under the adjoint action by g and into 
  the image ℓδ(g) 
  of g under the logarithmic derivative. We will mostly look at the two images 
  separately and then afterwards combine the results. This makes it possible to use the 
  root structure of G and 𝔤 to describe the gauge transformation.
   
  Proposition <ref> below shows that the image of an element of a linear algebraic group 
  under the logarithmic derivative lies in its Lie algebra.
  
  
  
  
  
   Let G ⊂GL_n be a linear algebraic group. Then the restriction of ℓδ to G
   maps G(F) to its Lie algebra 𝔤(F), that is
   
      ℓδ|_G: G(F) →𝔤(F)
   
  
  
   A proof can be found in <cit.>.
  
  In order to understand better the adjoint action we will use Remark <ref>
  below. It gives a description of the image of elements of a Chevalley basis under the adjoint 
  action of a root group element by the root system.
  
    For α, β∈Φ linearly independent let
   α - r β, … ,α + q β be the β-string through α 
   for r, q ∈ℕ and let ⟨α, β⟩ be the Cartan integer. We have
    
      Ad(u_β(x))(X_α)     =   ∑_i=0^q c_β, α,i x^i X_α + i β, 
      Ad(u_β(x))(H_α)    =    H_α - ⟨α, β⟩ x X_β , 
      Ad(u_β(x))(X_-β)    =    X_-β + x H_β - x^2  X_β
  
  where c_β,α,0=1 and c_β,α,i= ±r+ii.
  
   
   
  
  
  § CONNECTING THE STRUCTURE OF THE CLASSICAL GROUPS
   WITH PICARD-VESSIOT EXTENSIONS
  
  
  
  
  In this setion we will establish a connection between the geometrical structure of a 
  classical group G and a G-primitive Picard-Vessiot extension 
  (see Definition <ref>)
  of the differential field F. 
  This link will be obtained by applying the Bruhat decomposition to a fundamental 
  solution matrix of the extension. 
  In the following we denote by C[GL_n] the ring 
  C[X_ij,det(X_ij)^-1] in the indeterminates X_ij with 1 ≤ i,j ≤ n and
  for a linear algebraic group H ⊆GL_n we write C[H] for the coordinate ring of H, that is the quotient ring of C[GL_n] by the defining ideal of H.
  Further we denote by 
  X_ij the image of X_ij in C[H] and we write shortly X and X
  for the matrices (X_ij) and (X_ij) respectively.
  Finally, if H is connected we 
  mean by C(H) the field of fractions of the coordinate ring C[H].
  
  
  
  
  
  
  
  
  
  
  
  There are algebraically independent elements z=(z_1, …,z_l) and 
  y=(y_1, …,y_m) of C[B^-] such that 
  
      X =  t(z)
      u(y).
  
  
  
  
  We shortly write U for the direct product U_1 ×…× U_m and  
  analogously T for the direct product T_1 ×…× T_l.
  The product maps U→ U^- and T→ T are 
  isomorphisms of varieties and since T_i and U_i are isomorphic to 𝔾_m 
  and 𝔾_a, we identify the coordinate ring of T_i with 
  C[z̅_i,z̅_i^-1] and the coordinate ring of U_i 
  with C[y̅_i] for some new indeterminates z̅_i and y̅_i respectively. 
  Together we obtain a C-algebra isomorphism 
  
      C[T×U ] → 
      C[z̅_1,z̅_1^-1,…,z̅_l,z̅_l^-1 , y̅_1,…,  y̅_m]
  
  where z̅=(z̅_1, … ,z̅_l) and 
  y̅=(y̅_1, … ,y̅_m) are algebraically independent.
  From <cit.> we obtain that the product map from 
  T × U^- to B^- is an isomorphism and combined with the above product maps 
  it follows that the map 
  
      φ:T×U→ B^-, (t_1,…,t_l,u_1,…,u_m) ↦  t_1⋯  t_l   u_1 ⋯ u_m
  
  is an isomorphism of varieties. Thus its comorphism 
  
      φ^*: C[B^-] → C[T×U],  X_ij↦ (t(z̅)  u(y̅))_ij
  
  is an isomorphism of C-algebras and its inverse sends the algebraic independent elements 
  z̅ and y̅ to algebraic independent elements 
  z=(z_1, … ,z_l) and y=(y_1, … ,y_m) of the coordinate 
  ring of B^-.  Putting everything together we have  
  
      X=φ^*-1(t(z̅)  
      u(y̅)) 
      =t( z)  u( y)
  
  where z and y have the required property.
  
  
   
  There are algebraically independent elements x=(x_1, …,x_m),
  z=(z_1, …,z_l) and y=(y_1, …,y_m) of the 
  field of fractions C(G) of C[G] such that 
  
      X = u(x)n(w̅) t(z)
      u(y).
  
  
  
  Let U and T be as in the proof of Lemma <ref>. 
  We show that the map φ: U×T×U→ G 
  defined by
  
      (u_1,…,u_m,t_1,…,t_l,u̅_1,…,u̅_m)
      ↦ u_1⋯ u_m n(w̅) t_1 ⋯ t_l u̅_1 ⋯u̅_m
  
  is an isomorphism onto an open subset of G, that is, we prove that 
  U×T×U and G are birational equivalent. 
  The image of φ is the variety U^-n(w̅)B^- ⊂ G and we have to show 
  that it is an open subset of G. By <cit.> the subset 
  
      n(w̅)U^-n(w̅)B^-=U^+ B^-
  
  is open in G. It is the isomorphic image of U^-n(w̅)B^- under left multiplication with 
  n(w̅). Since left multiplication is a continuous morphism, we conclude that 
  U^-n(w̅)B^- is an open subset of G. Hence, φ is a birational morphism.
  
  As in the proof of Lemma <ref> we identify the coordinate ring 
  C[U×T×U] with 
  
      C[x̅_1,…,  x̅_m,z̅_1,z̅_1^-1,…,z̅_l,z̅_l^-1 , 
      y̅_1,…,  y̅_m]
  
  for algebraically independent elements x̅=(x̅_1, …, x̅_m), 
  z̅=(z̅_1, …, z̅_l) and 
  y̅=(y̅_1, …, y̅_m) which represent the coordinates of 
  the first, second and third factor of U×T×U 
  respectively. Since φ is a birational morphism, its comorphism 
  
      φ^* : C(G) → C(U×T×U),  X_ij↦
      (u(x̅)n(w̅)t(z̅)
      u(y̅))_ij
  
  is an isomorphism of fields. Its inverse sends the algebraically independent elements 
  x̅, z̅ and 
  x̅ to algebraically independent elements x=(x_1, …,x_m),
  z=(z_1, …,z_l) and y=(y_1, …,y_m) of C(G). 
  This yields  
  
      X=φ^*-1( u(x̅) n(w̅) 
      t(z̅) u(y̅))=
      u( x)n(w̅) t( z) u( y)
  
  with the required properties.
  
  
  
  
  
  Let 
   
      X= 
      u(x̅)n(w̅) t(z̅)
      u(y̅)
  
   be the normal form as in Lemma <ref>.
   
     
    *  Let g of G(C). 
       Then there are over C algebraically independent elements 
       x=(x_1, …,x_m), z=(z_1, …,z_l) and 
       y=(y_1, …,y_m) of C(G) such that 
       
      X g = u(x)n(w̅) t(z)
                u(y).
  
    
    *  Let i =0,…,l-1 and b ∈B̅^-_i(C). 
       Then there are over C(x̅,z̅_1,…,z̅_i) 
       algebraically independent elements  
       z_i+1,…,z_l and y=(y_1, …,y_m) of C(G) such that 
       
      X b = u(x̅)n(w̅) t_1(z̅_1) ⋯ 
            t_i(z̅_i)t_i+1(z_i+1)⋯ t_l(z_l) u(y).
  
    
    *  Let i =0,…,m-1 and u ∈U̅^-_i. 
    Then there are over 
    C(x̅,z̅,y̅_1,…,y̅_i) 
    algebraically independent elements y_i+1,…,y_m of C(G) such that 
       
      X u = u(x̅)n(w̅) 
            t(z̅)
            u_1(y̅_1) ⋯ u_i(y̅_i)u_i+1(y_i+1)⋯ u_m(y_m) .
  
  
  
  
   (a) Let g ∈ G(C). The translation map 
   ψ: G(C) → G(C),  x ↦ xg 
   is an isomorphism of varieties and so its comorphism is an C-isomorphism of fields
   
      ψ^*: C(G)→ C(G),  X_ij↦ (Xg)_ij.
  
   Applying ψ^* to the normal form of X we obtain 
   
      Xg = ψ^*(u(x̅)n(w̅) 
        t(z̅) u(y̅))=
        u(x)n(w̅) t(z)
        u(y)
  
  where the images x=(x_1, …,x_m), z=(z_1, …,z_l) and 
  y=(y_1, …,y_m) of x̅, z̅ and 
  y̅ under ψ^* are algebraically independent over C.
  
  (b) Let b ∈B̅^-_i(C) and represent it as a 
  product t_i+1⋯ t_l u with u ∈ U^-(C) and t_i+1∈ T_i+1(C), …, t_l ∈ T_l(C). 
  Using that U^-(C(G)) is normal in B̅^-_i(C(G)) we compute
  
      t(z̅)u(y̅) t_i+1⋯ t_l u =
      t(z̅)t_i+1⋯ t_l u(y) =  
      t_1(z̅_1) ⋯ 
      t_i(z̅_i)t_i+1(z_i+1)⋯ t_l(z_l)
      u(y)
  
  where z_i+1,… ,z_l and y=(y_1,…,y_m) 
  are elements of C(G). From the last equation we obtain the normal form 
  
      u(x̅)n(w̅) t(z̅)
      u(y̅) b = u(x̅)n(w̅)
      t_1(z̅_1) ⋯ t_i(z̅_i)t_i+1(z_i+1)⋯ t_l(z_l)
      u(y).
  
  Since the elements in the normal form are uniquely determined, 
  it follows with (a) that x̅,  
  (z̅_1,…,z̅_i, z_i+1,…,z_l)
  and y are algebraically independent over C.
  
  (c) For u ∈U̅^-_i(C) the subgroup structure 
  gives the identity  
  
      u(y̅)u = u_1(y̅_1) ⋯ 
      u_i(y̅_i)u_i+1(y_i+1)⋯ u_m(y_m)
  
  with y_i+1,… ,y_m in C(G). 
  The rest of the proof works as in (b).
  
  
  The next step is to connect our results with Picard-Vessiot theory. 
  We will mainly consider a specific type of Picard-Vessiot extensions. 
  
  
   A Picard-Vessiot extension E/F is called a full G-primitive extension of F if 
   the differential Galois group of E/F is G(C) and if there is a matrix Y ∈ G(E) 
   whose entries generate E over F and which satisfies 
   ℓδ (Y)=A ∈𝔤(F). 
  
  
   
  Let E/F be a full G-primitive Picard-Vessiot extension with matrix Y ∈ G(E).
   Then there are over F algebraically independent elements  
   x=(x_1, …, x_m), z=( z_1, … , z_l) and 
   y=(y_1, … ,y_m) of E such that 
   
      Y= u(x)   n(w̅) 
        t(z)  
        u(y) .
    
  
  
  The matrix A=ℓδ(Y)  ∈𝔤(F) is a defining 
  matrix for the Picard-Vessiot extension E of F. We use the standard construction method 
  to built a differentially isomorphic Picard-Vessiot extension for A. 
  Since A  ∈𝔤(F) and the differential 
  Galois group for A is G(C) the defining ideal of 
  G in C[GL_n ] extends to a maximal differential ideal of F[GL_n]. 
  Its quotient ring S is then a Picard-Vessiot ring  for A and the matrix 
  X∈ G(S) is by construction a fundamental solution matrix. 
  Denote by L the field of fractions of S. We have the embedding C(G) ↪ L 
  and so the normal form of X in G(C(G)) which we obtain from 
  Lemma <ref>
  yields the normal form of 
  the fundamental matrix X∈ G(L).
  
  Since E and L are both Picard-Vessiot extensions for A, 
  there is a differential F-algebra isomorphism 
  
      φ: L → E,  X↦ Y g
  
  where g ∈GL_n(C). Because the matrix X satisfies the defining 
  conditions for G, its image also does and so Yg lies in G(E). We conclude that g also lies 
  in G(C). Applying Corollary <ref> to 
  Xg^-1 we obtain that there are over C algebraically independent elements 
  x̅=(x̅_1, …,x̅_m), 
  z̅=(z̅_1, …,z̅_l) and 
  y̅=(y̅_1, …,y̅_m) of C(G) such that 
  
      X g^-1=u(x̅)n(w̅) 
      t(z̅) u(y̅).
  
  Since φ is an F-algebra isomorphism there are over F algebraically independent 
  elements x=(x_1, …,x_m), z=(z_1, …,z_l) and 
  y=(y_1, …,y_m) of E such that 
  
      Y= φ(u(x̅)n(w̅) 
      t(z̅) u(y̅))
      =u(x)n(w̅)t(z)
      u(y).
  
  
  
  
   For a full G-primitive extension E/F with fundamental matrix Y ∈ G(E) we call the 
   elements x=(x_1,…, x_m), z=(z_1,…, z_l) and 
   y=(y_1,…, y_m) of Proposition <ref> the coefficients of the normal form of Y. 
  
  
  The Borel subgroups, which are by definition maximal solvable subgroups, play 
  an important role in the structure theory of reductive groups. In differential Galois
  theory Picard-Vessiot extensions with solvable Galois group form a special class of 
  extensions. We will see now how these two concepts are linked in our setting.    
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Let x, z and y be the coefficients of a 
  normal form of full G-primitive extension E/F.
  Then L=F(x) is a differential field with constants C. 
  Moreover, E/L is a Picard-Vessiot extension with differential Galois group 
  B^-(C) and E is generated as a field by z and y over L. 
  Further E/L is a Liouvillian extension with tower of differential fields
  
      L(z,  y_1,…, y_m-1)(y_m)
       ⊃…⊃ L(z) (y_1) ⊃  
       L(z_1,…,z_l-1)(z_l) ⊃⋯⊃ 
       L(z_1) ⊃ L
  
  where the elements z are exponentials and the elements y 
  are integrals.
  
  
  
  Applying the Galois correspondence we obtain that E is a 
  Picard-Vessiot extension of the fixed field E^B^- with differential Galois group B^-(C). 
  Since B^- is solvable, the extension E/E^B^- is a Liouvillian 
  extension by <cit.>. The field E is generated 
  by z and y over F( x ), because we have 
  E=F( x, z,y).
  We show that E^B^-=F( x ). This will complete the proof of  
  the lemma except for the statement with the tower of fields. 
  
  It follows from Corollary <ref>, 
  applied with i=0, that the elements x are left invariant by all b ∈ B^- 
  and so the differential field E^B^- contains F ( x ).
  We prove that the two fields are actually equal by comparing their transcendence degrees.
   Because G is connected and of dimension 2m+l, the field extension E of F is a  
   transcendental extension of degree 2m+l. As an intermediate field E^B^- is also a transcendental extension of F.
   Since B^- is connected and 
   has dimension l+m, the extension E of E^B^- is a transcendental extension of degree l+m.
   We conclude that the transcendence degree of E^B^- over F is m. 
   Since x=(x_1, …, x_m) 
   are algebraically independent over F, the transcendence degree of F( x ) is 
   also m. It follows that E^B^-=F( x ) and so F( x) has the required properties.
   
   It is left to show that there is a tower of differential fields as stated and that the elements 
   z are exponentials and the elements y are integrals. 
  Following the notation of Chapter <ref> we have a 
  descending chain of subgroups as in (<ref>).
  
  For i=0, …, l-2 the quotients of B̅_i^- by B̅_i+1^- and the quotient 
  of B̅_l-1^- by U̅_0^- are isomorphic to 𝔾_m and so the fixed fields
  
      E^B̅_i+1^-⊃  E^B̅_i^-  and    E^U̅_0^-⊃ E^B̅_l-1^-
  
  are Picard-Vessiot extensions of transcendence degree one and they are generated by an exponential. 
  From above we already know that F(x) = E^B̅_0^-(C) and from 
  Corollary <ref> and <ref>, applied with i=0, we obtain for i=1, …, l-1 that 
  
      F(x, z_1, … , z_i) ⊆ E^B̅_i^-(C)  and   
      F(x, z_1, … , z_l) ⊆ E^U̅_0^-(C)
  
  One proves now inductively that the inclusions are actually equalities by comparing the transcendence 
  degrees of the corresponding extensions. Summing up we have that for i=1,…,l the extensions
  F(x, z_1, … , z_i-1) ⊂ F(x, z_1, … , z_i) are 
  Liouvillian and z_i is an exponential.
  
  For i=0,… m-1 we have that the quotient of U̅_i^- by U̅_i+1^- is isomorphic to 𝔾_a. 
  Thus the fixed fields 
  
      E^U̅_i+1^-⊃ E^U̅_i^-
  
  define Picard-Vessiot extensions of transcendence degree one which are obtained by adjoining an integral.
  From above we know that F(x, z_1, … , z_l) = E^U̅_0^-(C) and 
  trivially F(x, z, y_1, … , y_m) = E^U̅_m^-(C). Moreover, 
  Corollary <ref> 
  yields for i=1,… , m-1 that 
  
      F(x, z, y_1, … , y_i) ⊆ E^U̅_i^-(C).
  
  As above if we compare the transcendence degrees of the corresponding extensions we can show 
  that the inclusions are actually equalities. We conclude that for i=1,…,m the extension 
  F(x, z, y_1, … , y_i) of F(x, z, y_1, … , y_i-1) 
  is Liouvillian and that y_i is an integral.
  
  
  
  
  In the notation of Proposition <ref> the matrix 
  ℓδ (Y) ∈𝔤(F) is a defining matrix for the 
  Picard-Vessiot extension E of L. 
  It is actually very easy to produce a defining matrix for this extension 
  which lies in the Lie algebra 𝔟^-(L). 
  Indeed, the inverse of the matrix u(x)n(w̅) 
  is an element of G(L) and so the gauge transformation of ℓδ (Y) 
  with this matrix is defined over L and gives an equivalent defining matrix for the 
  extension. 
  The gauge equivalent defining matrix is the 
  logarithmic derivative of 
  
      (u(x)n(w̅))^-1 Y 
      =  t(z)u(y)
  
   which lies in 𝔟^- (L) by Proposition <ref>.
  
  
  
   
  
  
  
  § THE CONSTRUCTION OF THE GENERAL EXTENSION FIELD
  
  
  In this chapter we construct our  general extension field E. 
  It will be a Picard-Vessiot extension with differential Galois group G of
  a differential field which has differential transcendence degree l 
  over C. The differential structure of E will be associated to a 
  defining matrix of specific form. In <cit.> we 
  constructed a matrix differential equation depending on l parameters 
  t=(t_1, …, t_l) which defines a Picard-Vessiot extension 
  of the differential field generated over C by the differential indeterminates t
  with Galois group G(C). More precisely, for l 
  complementary roots γ_1, …,γ_l of Φ^- 
  
  and for A_0^+ the equation is defined by the matrix 
   
      A_G(t)=A_0^+ + ∑_i=1^l t_i X_γ_i .
  
  The construction of our general extension field will be based on the differential field 
  C⟨η⟩ where η=(η_1,…,η_l)
  are l differential indeterminates over C. It will be a Liouvillian extension of 
  C⟨η⟩ whose  differential Galois group will be B^-(C). 
  The defining matrix of the Liouvillian extension will be 
  the sum of a parametrization  of the Cartan subalgebra by η and 
  A_0^-. It will turn out that a fundamental matrix of this extension is 
  t(z) u(y) where 
  z=(z_1, …, z_l) are exponentials and 
  y=(y_1, …, y_l) are integrals. 
  Using the root structure we will show that there are m-l differential polynomials 
  
      f_l+1(η), …, f_m(η)
  
  of C{η} with the property that the logarithmic derivative of 
  
      u(η_1, …, η_l,
      f_l+1(η), …, f_m(η ) ) n(w̅) 
      t(z) u(y)
  
  is A_G(h) where 
  h=(h_1(η) , …, h_l(η)) are  
  elements of C{η}.
  Finally we will prove that h are differentially algebraically 
  independent over C and that E is a Picard-Vessiot extension of 
  C⟨h⟩ for A_G(h) 
  with differential Galois group G(C). 
  The degree of freedom of our general extension field E lies in the l 
  differential indeterminates η=(η_1,…,η_l).
  
  
   For technical reasons we start the construction of E with a tuple of m
   differential indeterminates η_m =(η_1, …, η_m)
   which we will reduce later to the tuple of the first l indeterminates 
   η =(η_1, …, η_l). To shorten notation 
   we will in the following denote by 
   u_i(η_m) the product 
   u_1(η_1) ⋯ u_i(η_i)  for i=2, … m-1.
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
   
  The image of u(η_m) 
  under the logarithmic derivative is 
   
      ℓδ( u(η_m))= ∑_i=1^l η_i'  X_i +
       ∑_i=l+1^m (η_i' + v_i(η_m)) X_i
  
   where v_i(η_m) lies in 
   C{η_1 , …,η_s_2} 
   with s_2 maximal such that r(s_2) =r(i)+1 and all its terms are of
   order one and of degree greater than one.
   
  
   
  With the product rule and the 
  definition of the adjoint action 
  we obtain for the image of u(η_m)
  under the logarithmic derivative the sum 
  
      ℓδ( u(η_m))  = 
      ℓδ(u_1(η_1))  + 
      Ad(u_1(η_1)) ( ℓδ(u_2(η_2))) + … +    
      Ad( u_m-1(η_m))(ℓδ(u_m(η_m))).
  
  We are going to determine the logarithmic derivatives ℓδ(u_i(η_i )) 
  for i=1,…,m and the images of ℓδ (u_k+1(η_k+1 )) under 
  Ad( u_k(η_m)) for k=1,…,m-1. 
  Using the results the statement of the lemma will follow from the last equation. 
   
  The logarithmic derivative of u_i(η_i) is η_i' X_i for i=1,…,m.
  Indeed, Proposition <ref> yields that ℓδ (u_i(η_i)) 
  lies in the root space 𝔤_i. 
  Moreover, the root group element u_i(η_i) is the image of η_i X_i  under the exponential map where X_i 
  nilpotent. Using this representation of u_i(η_i) 
  one easily checks that the only contribution to X_i in the corresponding
  product of power series is η_i' X_i.
  
  For j = 2, …, m let X be a linear combination of basis elements 
  X_i with r(i) ≤ r(j) whose the coefficients lie in C{η_1, …, η_j_2} with j_2 maximal such that 
  r(j_2)=r(j). We claim that for k = 1, …, j-1
  the image of X under the adjoint action of 
  u_k(η_k) can be represented as a linear combination of the same basis elements and that the coefficients of those X_i with r(i)=r(j) 
  are the same as in X and of those with r(i)<r(j) lie again in C{η_1, …, η_j_2}. Let  i be an index with r(i) ≤ r(j)
  and let v(η_m) be a nonzero element of 
  C{η_1, …, η_j_2}. 
  From the first formula in Remark <ref> we obtain
  
      Ad(u_k(η_k))( v(η_m) X_i )
      = ∑_s≥ 0  c_β_k,β_i,s η_k^s  v(η_m)   X_β_i + s β_k
  
  where the coefficient 
  c_β_k,β_i,s is nonzero
  if β_i + s β_k is a root and zero otherwise.
  Clearly if β_i + s β_k is a root, then it is equal to β_i (case s=0) or it is of height less than β_i (case s ≥ 1). Thus the right hand side of (<ref>) is a linear combination of 
  basis elements corresponding to roots of height less or equal than r(i). For s=0 the coefficient of X_i is v(η_m) and 
  because the index k is less than j,  
  the coefficients 
  in the sum of the right hand side of (<ref>)
  lie in C{η_1, …, η_j_2} for s ≥ 1. The claim now follows from the 
  linearity of the adjoint action.
  
  For k=1, …,m-1 we apply the claim iteratively to 
  
      Ad( u_k(η_m))(ℓδ (u_k+1(η_k+1))
      =Ad(u_1(η_1))( … ( Ad(u_k(η_k))(η_k+1' X_k+1)) … ).
  
  Representing the result as a linear combination of X_1,…,X_m
  we obtain that the coefficients of X_i with r(i) ≥ r(k+1)
   and i ≠ k+1 are zero, that the coefficient of   
  X_k+1 is η_k+1' and that the coefficients of X_i with r(i)<r(k+1) are in  C{η_1, …, η_s} where s is maximal such that r(s)=r(k+1). 
  In case of the latter coefficients we can adapt the definition of the ring 
  containing them such that it becomes independent of k. Let s_2 be maximal such that 
  r(s_2)=r(i)+1. For basis elements with r(i)+1= r(k+1) the ring 
  C{η_1, …, η_s_2} coincides with C{η_1, …, η_s} and for basis elements with r(i)+1< r(k+1) the ring 
  C{η_1, …, η_s_2} contains the ring C{η_1, …, η_s}.
  
  
  
  
  We consider the group SL_4(C).
  
  
  
  In <cit.> we worked out the 
  root system of type A_3 and presented a corresponding explicit Chevalley basis of the Lie algebra 𝔰𝔩_4(C). Our computations 
  will be with respect to this basis. We denote and number the negative roots  
  according to Chapter <ref> as 
  
      β_1 = - α̅_1 ,  β_2 = - α̅_2 ,  β_3 = - α̅_3 ,  β_4 = - α̅_1 - α̅_2,  β_5 = - α̅_2 - α̅_3 , 
      β_6 = - α̅_1 - α̅_2 - α̅_3
  
  so that r(i)=-1 for i=1,2,3, r(i)=-2 for i=4,5 and r(i)=-3 for i=6.
  We write for the basis element X_β_i and for a root 
  group element u_β_i shortly X_i and u_i respectively.
  From the exponential map we obtain that for a 
   negative root β_i the element u_i(η_i) of the root group U_i is the matrix 
   u_i(η_i)  = E + η_i X_i where E denotes 
   the 4× 4 unit matrix. It follows that ℓδ (u_i(η_i))  =  η_i' X_i.
  We have 
  
      ℓδ (u(η_6)) = η_1' X_1 +
           Ad(u_1(η_1))(η_2' X_2) + 
           Ad(u_1(η_1)u_2(η_2))(η_3' X_3) + 
      Ad(u_1(η_1)⋯ u_3(η_3))(η_4' X_4) +
           Ad(u_1(η_1)⋯ u_4(η_4))(η_5' X_5) + 
      Ad(u_1(η_1)⋯ u_5(η_5))(η_6' X_6)  .
  
  We determine the terms in the coefficients of the linear representation of the 
  logarithmic derivative of u(η_6) with respect to the 
  basis { X_i | i=1,…,6 }. Obviously for all
  i=1,…,6 the coefficient of X_i contains the term η_i'. Since 
  β_2 + β_1=β_4 the second summand contributes the term η_1η_2'
  to the coefficient of X_4. From the third summand the coefficients of 
  X_5 and X_6 obtain the terms η_2η_3' and η_1 η_2η_3' respectively, 
  since β_3 + β_2= β_5 and β_5 + β_1=β_6. The fourth summand 
  contributes the term η_3 η_4' to the coefficient of X_6, since 
  β_4 +β_3=β_6. From the next summand the coefficient of X_6 obtains the 
  term η_1η_5', since β_5 +β_1=β_6. Finally the last summand 
  gives no contribution, because the sum of β_6 and any negative root is not a root. 
  Putting our results together we obtain 
    
      ℓδ (u(η_6)) = 
        η_1' X_1 + η_2' X_2 + η_3' X_3  
       +  
       ( η_4' + v_4(η_6) ) X_4 + (η_5' + v_5(η_6)) X_5 +  
        (η_6' + v_6(η_6)) X_6
  
   with 
   
      v_4(η_6)= - η_2' η_1,  v_5(η_6)=-η_3' η_2,   v_6(η_6)= η_3 η_4' - η_5' η_1 + η_3' η_2 η_1
  
   and one easily checks that the coefficients are as in the statement of the lemma.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  We have  
   
      Ad(u(η_m))(A_0^+)=  
        A_0^+ + ∑_i=1^l g_i(η) H_i + ∑_i=1^m (ℓ_i(η_m)  + p_i(η_m) ) X_i
  
   where 
   
    
    *  g_1(η),…, g_l(η)  ∈ C[η_1, …,  η_l] are 
    nonzero C-linear independent homogeneous polynomials of degree one.
    
    *  p_i(η_m) ∈ C[η_1 ,…, η_i_2] with i_2  maximal such 
    that r(i_2)=r(i) and each term of p_i(η_m) is of degree greater than one.
    
    *  ℓ_i(η_m) ∈ C[η_k_1, … , η_k_2] 
    with k_1 minimal and k_2 maximal such that r(k_1)=r(i)-1=r(k_2)
    and ℓ_i(η_m) is a homogeneous polynomial of degree one. 
    Let i_1 be minimal and 
    i_2' maximal such that r(i_1)=r(i)=r(i_2') and i_2' does not 
    correspond to a complementary root. Then 
    
      ℓ_i_1(η_m)=0, …, ℓ_i_2'(η_m)=0
  
    is a quadratic linear system in η_k_1, … , η_k_2 of full rank.
  
  
  
  
  For j = 1, … , m we prove by induction that 
  
      Ad(u_j(η_m) )(A_0^+)=A_0^+ + ∑_i=1^j η_i W_i + 
      ∑_i=1^m p̅_i(η_m)  X_i
  
  where p̅_i(η_m) is zero or p̅_i(η_m) ∈ C[η_1, … , η_i_2] with i_2 maximal such that r(i_2)=r(i) and each term is 
  of degree greater than one.
  
  Since by definition W_j= [X_j,A_0^+] we have 
  [η_j X_j,A_0^+]=η_j W_j and from
  Remark <ref> we obtain  
  
      Ad(u_j(η_j))(A_0^+) = A_0^+ + η_j W_j + ∑_k=1^l ∑_s ≥ 2 c_β_j, α̅_k,s η_j^s  
       X_α̅_k + s β_j
  
  where the coefficients of the double sum are either zero or trivially a
  homogeneous polynomial of degree greater than one in η_j depending 
  whether α̅_k + s β_j is a root or not. 
  
  For j=1 the above observations yield  
  
      Ad(u_1(η_1)  )(A_0^+)=A_0^+ + η_1 W_1 + p̅_1(η_m) X_1
  
  where p̅_1(η_m) ∈ C[η_1] is a homogeneous polynomial 
  of degree two.  
  
  For j > 1 we obtain with the same observations that   
  
      Ad(u_j(η_m) )(A_0^+) =
      Ad(u_j-1(η_m)) 
      (A_0^+ + η_j W_j  +
       ∑_i=j_1^m  p̅_i(η_m) X_i)
  
  where we take j_1 minimal such that r(j_1)=r(j) and the polynomials  
  p̅_i(η_m) in C[η_j] are either zero or 
  homogeneous of degree greater than one.
  The polynomial p̅_i(η_m) is the coefficient of the 
  basis element which corresponds to the root β_i =α̅_k + s β_j with s ≥ 2 
  and so for i_2 maximal such that r(i_2)=r(i)
  we have p̅_i(η_m) ∈ C[η_1, …, η_i_2]. 
  Since the adjoint action is linear we can consider the three images
  on the right hand side of the last equation
  individually and then combine our results. For the image 
  of A_0^+ the induction assumption implies 
  
      Ad(u_j-1(η_m ))(A_0^+) = A_0^+ + ∑_i=1^j-1η_i W_i + 
      ∑_i=1^m p̅_i(η_m) X_i
  
  where p̅_i(η_m) is zero or p̅_i(η_m) ∈ C[η_1, … , η_i_2] with i_2 maximal such that r(i_2)=r(i) and each term of it is 
  of degree greater than one. 
  
  The vector η_j W_j is contained in the sum of root spaces
  corresponding to roots of height r(j)+1, that is in g^(r(j)+1), and  
  η_j lies in C [η_1, …, η_j_2] with j_2 maximal such that r(j_2)=r(j). 
  The image of η_j W_j computes with the above observations iteratively as
  
      Ad(u_j-1(η_m ))(η_j W_j ) = η_j W_j + 
      ∑_i=j_1^m p̅_i(η_m) X_i
  
  where j_1 is minimal such that r(j_1)=r(j). Since the entries of 
  u_j-1(η_m ) are in 
  C[η_1, …, η_j-1] and 
  η_j ∈ C [η_1, …, η_j_2] with j_2 maximal such that 
  r(j_2)=r(j), we conclude that all p̅_i(η_m) are 
  contained in C[η_1, …, η_j_2]. Because in each 
  iteration step s≥ 1 in the above formula and because we apply the adjoint action 
  to basis elements with polynomial coefficients whose terms are of degree at least one, the 
  degree of each term in p̅_i(η_m) is greater than one. 
  For i ≥ j_1 we have that i_2 
  maximal such that r(i_2)=r(i) satisfies i_2 ≥ j_2 and so p̅_i 
  also lies in the eventually larger ring C[η_1, …, η_i_2]. 
  
  For the last image the above observations yield 
  
      Ad(u_j-1(η_m ))
       (∑_i=j_1^m p̅_i(η_m) X_i)
       = ∑_i=j_1^m p̅_i(η_m)  X_i + 
       ∑_i=j_1'^m p̂_i(η_m) X_i
  
  where j_1' is minimal such that r(j_1')=r(j_1)-1. 
  For each i ≥ j_1' the index i_2 maximal such that r(i_2)=r(i) satisfies i_2 > j, 
  because j_1 is minimal such that r(j_1)=r(j). Since the entries of 
  u_j-1(η_m ) are elements of C[η_1, …, η_j-1] 
  and p̅_i(η_m) lies in C[η_j] and is of degree greater than one, 
  we conclude that p̂_i(η_m) is a polynomial of C[η_1, …, η_i_2] 
  where each term is of degree greater than one. Combining the three results completes the induction.
   
  The claim yields for j=m the equation 
  
      Ad(u (η_m) )(A_0^+)=A_0^+ + ∑_i=1^m η_i W_i + 
      ∑_i=1^m p̅_i(η_m) X_i
  
  where p̅_i(η_m) ∈ C[η_1, … , η_i_2] with 
  i_2 maximal such that r(i_2)=r(i) is a polynomial whose terms are 
  of degree greater than one.
  Since { W_i | r(i)=-1 } is a basis of 𝔥, we obtain
  that the coefficients of H_1,…, H_l 
  are nonzero C-linearly independent homogeneous polynomials 
  of degree one which we denote by g_i(η). 
  Let β_i be a non complementary root and let k_1 be minimal and 
  k_2 be maximal such that r(k_1)=r(i)-1=r(k_2). Since β_i and 
  at least one of the roots β_k_1, …, β_k_2 differ by a simple 
  root, the coefficient of X_i in (<ref>)
  has a non zero homogeneous part of degree one in 
  C[η_k_1,…, η_k_2 ]  stemming from W_k_1,…, W_k_2. 
  We denote this part by ℓ_i(η_m) and the remaining part of degree greater than one 
  by p_i(η_m). We prove that 
  
      ℓ_i_1(η_m)=0, …, ℓ_i_2'(η_m)=0
  
  with i_1 minimal and i_2' maximal such that 
  r(i_1)=r(i)=r(i_2') and  i_2' does not correspond to a complementary root 
  form a linear system in η_k_1,…,η_k_2 of full rank.
  Indeed, the set 
  { W_k_1,…, W_k_2}∪{ X_γ_k| r(k)=r(i) }
  is a basis of 𝔤^(r(i)) and so the coefficient matrix for 
  these vectors with respect to the basis X_i_1,…,X_i_2 with 
  i_2 maximal such that r(i)=r(i_2) has full rank. Since the two basis 
  coincide for the indices k_2 +1, …, i_2, it is a block matrix 
  whose last row consists of a corresponding zero and unit matrix. 
  We conclude that the
  first block, that is, the matrix without the rows and columns for the complementary roots has also full rank. 
  Identifying W_k_1,…, W_k_2 with η_k_1,…, η_k_2 shows that 
  
      ℓ_i_1(η_m)=0, …, ℓ_i_2'(η_m)=0
  
  is a linear system in η_k_1,…,η_k_2 of full rank.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  We proceed with Example <ref>. According to Lemma <ref> the image
  of the matrix A_0^+
  
  
  
  under the adjoint action with u(η_6) can be written as 
  
      Ad(u(η_6))(A_0^+) = 
         A_0^+ + ∑_i=1^3 g_i(η) H_i + ∑_i=1^6     (ℓ_i(η_6)+p_i(η_6) ) X_i .
  
  We determine the differential polynomials g_i(η), ℓ_i(η_6) and 
  p_i(η_6) ) and check that they have the described properties.
  We have
  
      Ad(u_6(η_6))(A_0^+)    =    A_0^+ + η_6 W_6 =  A_0^+ - η_6 X_4 + η_6 X_5  
      Ad(u_5(η_5))(A_0^+)    =    A_0^+ + η_5 W_5 =  A_0^+ - η_5 X_2 + η_5 X_3 
      Ad(u_4(η_4))(A_0^+)    =    A_0^+ + η_4 W_4 =  A_0^+ - η_4 X_1 + η_4 X_2
  
  Indeed, there are no nonlinear parts, since it is not possible to obtain a root by adding any simple root to 2 β_i for i=6,5,4. The components of W_i in the basis 
  { X_i | i=1,…,m } correspond to those roots which are the sum of β_i and 
  a simple root. Further with Remark <ref> we get 
  
      Ad(u_3(η_3))(A_0^+)    =    A_0^+ + η_3 W_3 - η_3^2 X_3  
      Ad(u_2(η_2))(A_0^+)    =    A_0^+ + η_2 W_2 - η_2^2 X_2 
      Ad(u_1(η_1))(A_0^+)    =    A_0^+ + η_1 W_1 - η_1^2 X_1
  
  where W_i=-H_i for i=3,2,1. The W_j with j=1,…,6 determine the terms in the  
  homogeneous polynomials of degree one g_i(η) and 
  ℓ_i(η_6). We have  
  
      g_1(η) = -η_1,  g_2(η)=-η_2 ,  g_3(η)=-η_3
  
  and they clearly satisfy Lemma <ref>. Further we read off 
  
      ℓ_1(η_6) = -η_4 ,  ℓ_2(η_6) = η_4 -η_5,  ℓ_3(η_6) = η_5,  
      ℓ_4(η_6) = -η_6,  ℓ_5(η_6) = η_6 ,  ℓ_6(η_6)=0
  
  and they obviously satisfy the first part of statement <ref>.
  Since the complementary roots are 
  β_3, β_5 and β_6 (see <cit.>), we have the two linear systems 
  
      ℓ_1(η_6) =0,  ℓ_2(η_6)=0   and  ℓ_4(η_6)=0
  
  in the variables η_4, η_5 and 
  in the variable η_6 respectively and both systems have full rank. 
  In order to determine the non-linear parts p_i(η_6) 
  one can use the recursion 
  
      r_0     :=   Ad(u_6(η_6))(A_0^+) 
      
       r_k   :=   Ad(u_6-k(η_6-k))(r_k-1)  for k=1,…,5
  
  which computes the whole image of A_0^+ under 
  Ad(u(η_6)). 
  We sketch how to proceed. We already know that in the steps k=3,4,5 we 
  obtain from A_0^+ in r_k-1 the term η_6-k^2 in the coefficient 
  of X_6-k in the linear representation of r_k. We get further nonlinear terms 
  from the component in the 
  Cartan subalgebra. Since it is only nonzero in the steps 4 and 5, we obtain 
  the new nonlinear terms η_2η_3 and η_2η_1 in the coefficients of 
  X_2 and X_1 in the linear representation of r_4 and r_5 respectively.
  Finally in each step k one obtains new 
  nonlinear terms in the coefficient of X_j in the representation of r_k 
  from multiplying the coefficient of
  that X_i in the representation of r_k-1 with η_6-k for which β_i + β_6-k is the root β_j.
  One can check that the nonlinear parts compute as 
  
      p_1(η_6) = -η_1^2 +η_2 η_1 ,   
      p_2(η_6) = -η_2^2+η_3 η_2 ,  
      p_3(η_6) = -η_3^2  , 
      
      p_4(η_6) = -η_2 η_4 + η_1 (η_2^2 -η_4  -η_3  η_2+η_5 ) ,  
      p_5(η_6) = -η_5 η_2+η_3 ( η_4- η_5+η_2 η_3 ) , 
      
      p_6(η_6) = -η_1 η_3 η_4-η_1 η_6-η_5 η_4+η_5 η_2 η_1-η_3^2   
          η_2 η_1+η_3 η_5 η_1-η_3 η_6
  
  and that they have the stated properties of Lemma <ref>.
  
  
  
  
  
  In the next step we construct the Liouvillian extension of C⟨η⟩. For this purpose we fix a representative 
  n(w̅) in the normalizer of the torus for the Weyl group element w̅ of maximal length. Since w̅ sends {- α̅_1, …, -α̅_l } bijectively to Δ and
  the adjoint action of n(w̅) maps X_1, …, X_l bijectively to non zero multiples of
  X_α̅_1, …, X_α̅_l, there are units 
  c=(c_1,…,c_l) in C such that 
  
      Ad(n(w̅))(A_0^-(c)) =A_0^+.
  
  Further the adjoint action of n(w̅) sends each basis element H_i 
  of the Cartan subalgebra to a nonzero multiple. Thus for g_i(η) of Lemma <ref> there are nonzero C-linear 
  independent homogeneous polynomials g̅_1(η) ,…, g̅_l(η) 
  of degree one in C[η] such that 
  
      Ad(n(w̅))( ∑_i=1^l g̅_i(η) H_i ) = 
       ∑_i=1^l -g_i(η) H_i.
  
  With these definitions we set 
  
      A_L(η)= ∑_i=1^l  g̅_i(η) H_i +A_0^-(c) 
       ∈𝔟^-(C⟨η⟩)
  
  and consider the matrix differential 
  equation defined by A_L(η).
  
  
  There is a Picard-Vessiot extension E of C ⟨η⟩ for 
  A_L(η) with the following properties:
  
      
    *  The differential Galois group of E over 
      C ⟨η⟩ is B^-(C).
      
    *  There is a fundamental solution matrix Y_L ∈ B^-(E) and 
      there are elements z=(z_1,…,z_l) and y=(y_1,…,y_m) of E
      algebraically independent over C ⟨η⟩ with 
      
      Y_L= t(z) u(y).
  
      
    *  The extension is Liouvillian and there is a tower 
      of differential fields 
      
      E_0=C ⟨η⟩⊂ E_1 ⊂…⊂ E_l ⊂ E_l+1⊂…⊂ E_m+l =E
  
      where E_i = E_i-1(z_i) with z_i an exponential for i=1, …,l and E_i = E_i-1(y_i-l) 
      with y_i-l an integral for i=l+1, … , m+l. 
      
    *  The elements z and y have shape
    
      z_1= e^∫g̅_1(η), …,  z_l=e^∫g̅_l(η) , 
       
         y_1 = ∫ c_1 α̅_1 (t(z))^-1,  … ,  
         y_l= ∫ c_l α̅_l (t(z))^-1, 
      
         y_l+1 = ∫ -v_l+1(y),  …,  
         y_m = ∫ -v_m(y  ) ) ,
  
      where g̅_i(η) are as in the definition of 
      A_L(η) and v_l+1(y),… , v_m(y) 
      are as in Lemma <ref>.
  
  
  
  
  
  (a) We show that for n ∈ℕ there are nonzero elements 
  c=(c̅_1,… , c̅_l) of C such that the differential Galois group of a Picard-Vessiot extension 
  for the matrix  
  
      A_L=∑_i=1^l c̅_i z^n H_i + c_i X_i
  
  over the rational function field C(z) with standard derivation is the group B^-(C).
  The statement then follows from <cit.> and <cit.>.
  Indeed, since g̅_1(η), …, g̅_l(η) are C-linearly independent homogeneous polynomials of 
  degree one in C[η_1, …, η_l], the corresponding coefficient 
  matrix is invertible and so there are nonzero elements  
  ĉ_1, … , ĉ_l of C such that 
  
      c̅_i z^n = g̅_i (ĉ_1 z^n, … , ĉ_l z^n).
  
  It follows that the differential C-Algebra homomorphism σ:C{η}→ C[z] 
  defined by σ(η_i)=ĉ_i z^n satisfies σ(A_L(η))=A_L and is 
  a surjective R_1-specialization.
  
  In order to show that there exists c such that A_L has Galois group B^-(C) 
  we consider the quotient homomorphism 
  
      π : B^- → B^- / [U^-,U^-]
  
  and the corresponding Lie algebra homomorphism 
  d π : 𝔟^- →𝔟^- / [ 𝔲^-,𝔲^- ].
  According to <cit.> it is enough to prove that there exists c
  such that the differential Galois group of the image d π(A_L) 
  is the full group π(B^-). The first  
  assumption of the proposition is satisfied by <cit.>.
  For the remaining assumptions the standard construction method yields a Picard-Vessiot 
  extension E of F with a fundamental solution matrix Y ∈ B^-(E), 
  since A_L lies in 𝔟^-. 
  By <cit.> the logarithmic derivative of 
  π(Y) is d π(A_L) and so π(Y)
  is a fundamental solution matrix for 
  d π(A_L) and its entries generate a Picard-Vessiot extension. 
  The last condition is satisfied by construction.
  
  
  A basis of the Lie algebra dπ (𝔟^-) is given by 
  the images H_1, …, H_l of the basis H_1, …, H_l of the 
  Cartan subalgebra and the images X_1, …,X_l of the basis 
  elements X_1, …, X_l corresponding to the negative simple roots and so  
  
      dπ (A_L)= ∑_i=1^l c̅_i z^n H_i  + c_i X_i = A_L .
  
  We follow the argumentation of <cit.> and use the notation.
  We need to show that there are c such that 
  the images of 
  
      c_z=(c_1 z^n,…,c_l z^n)
  
  in the quotient 
  C (z)/ ℓδ(C (z)^*) are linearly independent over ℤ and that c_i does not 
  reduce to zero modulo L_c_z, α̅_i(C (z)). Indeed, if these statements are fulfilled, it follows then from <cit.> together 
  with <cit.> and <cit.> that the 
  differential Galois group of A_L is π(B^-).
  
  For the first statement <cit.> yields that for z^n there are infinitely many choices of 
  coefficients c in C such that c_z are 
  ℤ-linearly independent modulo ℓδ ( C(z)^*). 
  For the second statement we need to show for i=1,…,l that the differential 
  equations 
  
      L_c_z, α̅_i(ζ)= ζ' - ∑_j=1^l ⟨α̅_j, α̅_i ⟩c_j z^n ζ =c_i.
  
  have no solutions in C(z). 
  According to <cit.> for any nonzero c̅∈ C and 
  n ∈ℕ the differential Galois group of a Picard-Vessiot extension of C(z) 
  for the differential equation defined by the matrix 
  
      [                          0                          1; c̅^2 z^2n/4 + c̅ n z^n-1/2                          0 ]
  
  is conjugate to the Borel subgroup of SL_2. It can be easily checked that 
  this matrix is gauge equivalent to 
  
      [  c̅ z^n/2       c_i;         0 -c̅ z^n/2 ]
  
  and so there are no solutions in C(z) for differential equations of shape 
  
      ζ' -c̅ z^n ζ =c_i.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  (b) We construct a Picard-Vessiot extension E of C ⟨η⟩ for A_L(η) such that there is a 
  fundamental solution matrix Y_L in B^-(E) and elements 
  z=(z_1, …, z_l) and y=(y_1,…, y_m) of E 
  such that Y_L can be written as a product as stated where we use the notation of 
  Chapter <ref>. 
  Since A_L(η) lies in the Lie algebra of B^- the defining ideal of B^- in C[GL_n] extends to a differential 
  ideal in C ⟨η⟩[GL_n] where the derivation 
  on X is defined by 
  multiplication with A_L(η). 
  It follows from (a)   
  that it is a maximal differential ideal and so the quotient ring is a Picard-Vessiot 
  ring R. By construction the matrix Y_L:=  X is a 
  fundamental solution matrix and lies in B^-(R). Since C[B^-] embeds canonically 
  into R, Lemma <ref> yields that there are over C ⟨η⟩ algebraically independent elements 
  z=(z_1,…, z_l) and y=(y_1,…,y_m) of R such that 
  
      Y_L= t(z)  u(y).
  
  The field of fractions E of R has then the desired properties. 
  
  
  (c) The extension is clearly a Liouvillian extension, since its 
  differential Galois group is solvable. We prove that there is a tower of 
  fields as stated for the elements z and  
  y of (b) and that they are respectively exponentials and 
  integrals. The proof is similar to the one of Proposition <ref>. 
  Recall from Chapter <ref> that we have 
  a descending chain of normal subgroups for B^- as in 
  (<ref>). For i=0, …, l-2 the quotients 
  of B̅^-_i by B̅^-_i+1 and B̅^-_l-1 by U̅^-_0 are isomorphic to 
  G_m. Hence the corresponding inclusions of fixed fields 
  
      E^B̅^-_i+1⊃ E^B̅^-_i  and   
      E^U̅^-_0⊃ E^B̅^-_l-1
  
  are Picard-Vessiot extensions of transcendence degree one and they are 
  generated by an exponential. Corollary <ref> and <ref> (with i=0) implies after multiplication with the inverse
   of u(x)n(w̅) for i=1, …, l-1
  the inclusions 
  
      C⟨η⟩ (z_1, …, z_i) ⊆ E^B̅^-_i  and   
      C⟨η⟩ (z_1, …, z_l) ⊆ E^U̅^-_0.
  
  We can prove inductively by comparing the 
  transcendence degree of the respective fields that 
  
      E^B̅^-_i= E^B̅^-_i-1(z_i)   and   
      E^U̅^-_0= E^B̅^-_l-1(z_l)
  
  and it follows with the above that z_i an exponential.
  
  For i=0,…, m-1 the quotients U̅^-_i by U̅^-_i+1 are 
  isomorphic to 𝔾_a and so the corresponding inclusions 
  of fixed fields 
  
      E^U̅^-_i+1⊃ E^U̅^-_i
  
  are Picard-Vessiot extensions of transcendence degree one and the extensions 
  are generated by an integral. We already know that 
  E^U̅^-_0= C⟨η⟩ (z)
  and that E^U̅^-_m= C⟨η⟩ (z,y). For i=1, …, m-1 we conclude
  as above with Corollary <ref> that
  z,y_1, …, y_i are left fixed by U̅^-_i, that is we have
  
      C⟨η⟩ (z,y_1, …, y_i)
      ⊆ E^U̅^-_i.
  
  Again one proves inductively by comparing the transcendence degrees of the 
  respective fields that 
  the inclusions are actually equalities. Summing up it follows that 
  E^U̅^-_i= E^U̅^-_i-1(y_i) with y_i an integral.
  
  
  
  (d) We determine the shape of the exponentials z and the 
  integrals y. Using the product rule  
  the logarithmic derivative of the fundamental matrix 
  Y_L=t(z) u(y)
  computes as
  
      ℓδ  (Y_L) = ℓδ (t(z) ) + 
       Ad(t(z) ) 
      (ℓδ (u(y) ) ) =A_L.
  
  By Proposition <ref> the logarithmic derivative of 
  t(z) and u(y)
  are elements of 𝔥 and 𝔲^- respectively.
  Moreover the adjoint action of the torus stabilizes the root spaces and 
  so (<ref>) implies 
  
      ℓδ (t(z))= ∑_i=1^l g̅_i(η)  H_i.
  
  The logarithmic derivative of t_i(z_i) computes as z_i'/z_i  H_i. 
  Moreover with the product rule and the fact that the adjoint action 
  of the torus stabilizes the Cartan algebra we conclude that 
  g̅_i(η)=z_i'/z_i and so we have 
  
      z_i= e^∫g̅_i(η) .
   
  It is left to determine the integrals. From (<ref>) 
  we know that 
  
      Ad(t(z) ) 
      (ℓδ (u(y) ) ) =A^-_0(c).
  
  Since the torus acts on X_i by the roots we conclude with 
  Lemma <ref> that the 
  coefficients of basis elements corresponding to the negative simple roots 
  satisfy c_i= α̅_i( t(z)) y_i'. Thus for 
  i=1, …, l the integral y_i is 
  
      y_i = ∫ c_i α̅_i(t(z))^-1.
  
  For the remaining indices that is for the roots β_i with 
  r(i)≤ -2 the component of the 
  logarithmic derivative of u(y) in 
  𝔤_i has to be zero. It follows then from Lemma <ref> that for  l+1≤ i ≤ m we have 
  
      y_i = ∫ -v_i(y)
   
  where v_i(y) only depends on integrals y_1,…,y_j with j maximal such 
  that r(j)=r(i)+1.
  
  
  
  We continue with the example for SL_4(C). 
  For the Weyl group element of maximal length we choose the representative 
  
      n(w̅) = E_14 - E_23 + E_32 - E_41
  
  in the normalizer of the torus of SL_4(C). 
  With g̅_1(η)= - η_3, g̅_2(η)=- η_2, 
  g̅_3(η)=- η_1 and c_i= -1
  the defining matrix of the Liouvillian extension becomes
  
      A_L(η) = -A_0^- - η_3 H_1 - η_2 H_2 -η_1 H_3
  
  and it satisfies 
  
      Ad(n(w̅))(A_L(η)) = A_0^+ -g_1(η) H_1 - g_2(η) H_2 - g_3(η) H_3
  
  where g_i(η) is as in Example <ref>.
  With g̅_i(η) we define the exponentials 
  
      z_1 =  e^∫ - η_3,  z_2 = e^∫ - η_2,  z_3 = e^∫ - η_1
  
  and obtain for the roots of height -1 the integrals
  
      y_1 = - ∫ e^∫ (-2η_3 + η_2), y_2 = - ∫ e^∫ (-2η_2 + η_1 + η_3), y_3 = - ∫ e^∫ (-2 η_1 +  η_2) .
  
    Furthermore with v_i(η_6)
  as in Example <ref> we compute for the remaining roots the integrals   
   
      y_4    =   ∫ -v_4(y_1,y_2,y_3) = ∫(∫ e^∫( -2 η_3 + η_2)) e^∫ (-2 η_2 + η_1 + η_3) , 
      
           y_5    =   ∫ -v_5(y_1,y_2,y_3) = ∫( ∫ e^∫ (-2 η_2 +η_1+η_3)) e^∫ (-2 η_1+ η_2) , 
      
           y_6    =   ∫ -v_6(y_1,…,y_5) = ∫(∫ e^∫ (-2 η_1 + η_2)) (∫ e^∫ ( -2 η_3 +η_2)) e^∫ (-2 η_2 +η_1 +η_3).
  
  By construction  
  Y_L=t(z) u(y) satisfies 
  ℓδ(Y_L)=A_L(η).
  
   
  
  Let z,  y be as in Proposition <ref> 
  and define 
  
      Y= u(η_m)  n(w̅) 
       t(z) u(y).
  
  We have 
  
      ℓδ(Y)= A_0^+ + h_1(η_m) X_1 + … + h_m(η_m) X_m
  
  with coefficients
  
      h_i(η_m) = η_i'+ ℓ_i(η_m)  +  q_i(η_m)
  
   where ℓ_i(η_m) is as in Lemma <ref>
  and q_i(η_m) lies in C {η_1 ,… , η_s_2}[η_s_2 +1,…η_i_2] with s_2 and i_2 maximal such that r(s_2)=r(i)+1 and r(i_2)=r(i). Moreover each 
  term of q_i(η_m) is of degree greater than one and the derivatives appearing are at most of order one.
  
  
  
  With the product rule and ℓδ(n(w̅))=0 the logarithmic derivative of Y computes as
   
      ℓδ(Y) = ℓδ(u(η_m) ) + 
         u(η_m) n(w̅)  ℓδ(t(z) u(y))  
         ( u(η_m)n(w̅))^-1 .
   
  We look at the two summands individually and then combine our results. 
  From Lemma <ref> we obtain that the first summand 
  is an element of 𝔲^-(C{η}) and that if 
  we represent it as a linear combination of the basis elements X_1, … , X_m then 
  the coefficient of X_i is η_i'+v_i(η_m) where 
  v_i(η_m) ∈ C {η_1 , … , η_s_2} with s_2 maximal such that 
  r(s_2)=r(i)+1 and all its terms are of order one and of degree greater than one. 
  
  Next we consider the second summand.  
  Since z and y are as in Proposition <ref> we have 
  
      ℓδ( t(z) u(y))= A_L(η)=
      A_0^-(c) + ∑_i=1^l g̅_i(η)  H_i .
  
  The adjoint action of n(w̅) maps A_L(η) by construction to  
  
      A_0^+  + ∑_i=1^l -g_i(η)  H_i
  
  where g_i(η) is as in Lemma <ref>. 
  Moreover the linearity of the adjoint action yields 
  
      Ad( u(η_m)) (A_0^+) + 
      ∑_i=1^l - g_i(η) Ad( u(η_m))(H_i)   .
  
  It follows from Lemma <ref> that the first summand of 
  (<ref>) computes as 
  
      Ad( u(η_m))(A_0^+) = A_0^+ + 
        ∑_i=1^l g_i(η)  H_i + ∑_i=1^m ( ℓ_i(η_m)  + p_i(η_m) ) X_i
  
   where g_i(η)  ∈ C[η_1, …,  η_l] are nonzero C-linear 
   independent homogeneous polynomials of degree one, 
   ℓ_i(η_m) ∈ C[η_k_1, … , η_k_2]  with k_1 minimal and k_2 
   maximal such that r(k_1)=r(i)-1=r(k_2) and ℓ_i(η_m) is a homogeneous 
   polynomial of degree one and p_i(η_m)  ∈ C[η_1 ,…, η_i_2] with
   i_2 maximal such that r(i_2)=r(i) and each term of p_i(η_m) is of degree greater 
   than one. By the second formula of Remark <ref> 
   the adjoint action of u_k(η_k) maps H_i to 
   
      H_i + ⟨α̅_i, β_k ⟩ η_k   X_k
  
   and so the second summand of (<ref>) becomes
  
      ∑_i=1^l - g_i(η) Ad(u(η_m))(H_i) = 
       ∑_i=1^l - g_i(η)  H_i  + ∑_i=1^m a_i(η_m)  X_i
  
   where a_i(η_m) is a polynomial of C[η_1, …, η_i_2] with i_2 maximal 
   such that r(i_2)=r(i) and each term of a_i(η_m) is of degree greater than one.
  Indeed, this follows from the fact that a_i(η_m) is the product of 
  η_i ∈ C[η_1, …, η_i_2] 
  and the homogeneous polynomial of degree one 
  g_i(η)  ∈ C[η_1, …, η_l].
  
  Combining the results one checks that the coefficient of each H_i in the linear combination of ℓδ(Y) 
  vanishes and that the coefficient of each X_i is as described in the statement.
  
  
  
  We proceed with the example for SL_4(C). 
  The logarithmic derivative of 
  
      Y= u(η_6)n(w̅) t(z) u(y),
  
  where z and y are as in 
  Example <ref>, computes as 
  
      ℓδ (Y)    =   ℓδ (u(η_6)) + 
      Ad(u(η_6) n(w̅)) (ℓδ (t(z) u(y))) 
         =   ℓδ (u(η_6)) + 
      Ad(u(η_6)) ( A_0^+ - g_1(η) H_1 - g_2(η) H_2 -g_3(η) H_3 ).
  
  Combining the results of Example <ref> and 
  Example <ref> we obtain 
  
      ℓδ (Y) = A_0^+ + h_1(η_6) X_1 + ⋯ + h_6(η_6) X_6
  
  with coefficients 
  
      h_1(η_6)  =η_1' - η_4  + q_1(η_6) ,          
      h_2(η_6)  =η_2' + η_4 - η_5 + q_2(η_6),  
          h_3(η_6) = η_3' +  η_5 + q_3(η_6),        
      h_4(η_6) = η_4' - η_6  +q_4(η_6), 
          h_5(η_6) = η_5' + η_6 + q_5(η_6),       
      h_6(η_6) =  η_6' + q_6(η_6),
  
  where 
  
      q_i(η_6)= v_i(η_6) + p_i(η_6) +a_i(η_6)
  
  and a_i(η_6) is as in the proof of 
  Lemma <ref>. By computation we obtain 
  
      q_1(η_6) = η_1^2 , q_2(η_6) =   η_2(η_2 - η_1),   q_3(η_6) =   η_3 ( η_3 - η_2)
  
  and 
  
      q_4(η_6)    =    η_5 η_1  - ( η_2 
      (η_2 - η_1)) η_1 - η_3η_4     + v_4(η_6), 
      
      q_5(η_6)    =   η_3 η_4 - η_5 η_1  
      - ( η_3 (η_3 -η_2  )) η_2  + v_5(η_6),
      
      q_6(η_6)    =    η_3^2 (η_1 η_2 - η_4 )
      + η_3 ( η_4 η_2-  η_1 η_2^2 ) +η_5 ( η_1^2 - η_4) + v_6(η_6).
  
    Recall that the indices 1, 2, 3 correspond to the roots of height -1, 
    the indices 4, 5 belong to the roots of height -2 and the root belonging to the index 
    6 is the only root of height -3. It is easy to check that the linear and non-linear 
    parts of the coefficients satisfy the statement of the lemma.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Let 
  
      { i_1, …, i_m-l|β_i_s∉{γ_1, …, γ_l  }}∪{ j_1, …, j_l |β_j_s∈{γ_1, …, γ_l  }}
  
  be the partition of {1, …, m } into indices corresponding to non-complementary and complementary roots and let 
  h_1(η_m), …, h_m(η_m) be the 
  differential polynomials of Lemma <ref>. 
  Then the system of equations 
   
      h_i_1(η_m) =0, …, h_i_m-l(η_m) =0
   
   is equivalent to the system 
   
      η_l+1 = ℓ̅_l+1(η) + p̅_l+1(η), …, η_m = ℓ̅_m(η)  +  p̅_m(η)
  
   where ℓ̅_i(η) and p̅_i(η) have the following properties: 
    
       
    *  The differential polynomial 
       ℓ̅_i(η) ∈ C{η_1, …,η_j} is homogeneous of degree one 
       and of order |r(i)+1| with j maximal such that r(j)=-1 and j does not
       correspond to a complementary root.
       
    *  For the heights q=-2, …, r(m) let i_1 be minimal and i_2 
       be maximal such that r(i_1)=q=r(i_2). Then the coefficient matrix of the linear system 
       
      ℓ̅_i_1(η)=0,  …  ,ℓ̅_i_2(η)=0
  
       in the variables η_1^(|q+1|), …,η_j^(|q+1|) 
       has full rank. In case q ≤ -3 the linear system defined by this coefficient matrix is equivalent 
       to a subsystem obtained form the corresponding coefficient matrix for q+1 by canceling those 
       rows whose indices belong to complementary roots.
       
    *  Each term in the differential polynomial p̅_i(η) ∈ C{η} is of degree greater than two and the derivatives appearing are at most of order |r(i)+2|.
   
  
  
  
  
  We fix the following notation. For a height q=-1,…, r(m)
  let i_1 be minimal and i_2 and i_2' be maximal such that 
  r(i_1)=q=r(i_2)=r(i_2') and the index i_2' does not correspond to a 
  complementary root. Further let k_1 be minimal and k_2 be maximal 
  such that r(k_1)=q-1=r(k_2). If not otherwise stated the index i runs between i_1 ≤ i ≤ i_2'
  and the index k between k_1≤ k ≤ k_2.
  
  We use induction on the height q=-1,…, r(m). Let q=-1.
  Note that i_1=1, i_2'=j with j as in (a) and k_1 = l+1.
  By assumption  we have 
  
      h_1(η_m) = η_1'+ ℓ_i(η_m)  +  
        q_1(η), …, h_i_2'(η_m) = η_i_2''+ 
        ℓ_i_2'(η_m)  +  q_i_2'(η)
  
   with ℓ_i(η_m)  ∈ C[η_l+1, … , η_k_2] homogeneous of degree one 
   and q_i(η) ∈ C [ η_1 ,… , η_l ] 
   such that each term is of degree greater than one.
   Setting h_i (η_m)
   to zero and then solving for ℓ_i(η_m) we obtain the system 
    
      ℓ_1(η_m)  = - η_1' - q_1(η),  …,  
         ℓ_i_2'(η_m)  = - η_i_2'' - q_i_2'(η)
  
   where by assumption the left hand sides of the system 
   define a quadratic linear system 
   
      ℓ_1(η_m)=0,…,ℓ_i_2'(η_m)=0
  
   of full rank in the variables 
   η_l+1,…, η_k_2. Note that the right hand sides depend only on the variables
   η_1,…, η_l and the first order derivatives of η_1,…, η_i'_2.
   Applying Gaussian elimination we obtain the equations 
   
      η_l+1 = ℓ̅_l+1(η) + p̅_l+1(η),  …, η_k_2 = ℓ̅_k_2(η) + p̅_k_2(η)
  
   where ℓ̅_k(η) and p̅_k(η) represent the corresponding results obtained from applying the 
   same row operations to -η_i' and to q_i(η) 
   respectively. Clearly ℓ̅_k(η) satisfies statement (a) and 
   
      ℓ̅_l+1(η)=0, …, ℓ̅_k_2(η)=0
  
   is a linear system of full rank in η_1',… , η_i'_2'. Since the properties of 
   q_i also hold for their C-linear combinations, we conclude that
   p̅_k(η) satisfies (c).  
   
  Let q ≤ -2. 
  By the induction assumption we have the equations 
  
      η_l+1 = ℓ̅_l+1(η) + p̅_l+1(η),  …, 
       η_i_1-1 = ℓ̅_i_1 -1(η) + p̅_i_1-1(η), 
      η_i_1 = ℓ̅_i_1(η) + p̅_i_1(η),  …, 
       η_i_2 = ℓ̅_i_2(η) + p̅_i_2(η) ,
  
  
  
  
  
  
  
  
  
  
  
  where ℓ̅_i(η) and p̅_i(η) with i=l+1,…,i_2 satisfy (a), (b) and (c) respectively.
  We obtained these equations from solving the equations   
  h_i(η_m)=0 with i= 1, …, i_1-1 and i does not correspond 
  to a complementary root. Thus we did not yet use the differential polynomials
  
      h_i_1(η_m), …,h_i'_2(η_m) .
  
   Setting then to zero and solving for ℓ_i(η_m)
   we obtain the system of equations
   
      ℓ_i_1(η_m) =  - η_i_1' - q_i_1(η_m),  …,    
         ℓ_i'_2(η_m)  =  - η_i'_2' - q_i'_2(η_m)
  
   where by assumption the left hand sides define a quadratic linear system
  
  
  
   of full rank in the indeterminates η_k_1, … , η_k_2 and the right hand sides 
   depend on the differential polynomials 
   
      q_i(η_m) ∈ 
       C {η_1 ,… , η_i_1 -1}[η_i_1,…, η_i_2]
  
   whose terms are of degree greater than one and only contain derivatives of order at most  
   one. Substituting (<ref>) into - η_i_1', …,- η_i'_2'
   as well as (<ref>) and (<ref>) into 
   - q_i(η_m) the equations in
   (<ref>) become
    
      ℓ_i_1(η_m) =  - ℓ̅_i_1(η)' - q̅_i_1(η),  …,   
         ℓ_i'_2(η_m)  =  - ℓ̅_i'_2(η)' - q̅_i'_2(η)
  
   where in q̅_i(η) we collected the terms stemming from -p̅_i(η)' and -q_i(η_m). 
   Since the linear system 
   
      ℓ̅_i_1(η)=0, …,ℓ̅_i_2(η)=0
  
   in the variables η_1^(|q+1|), …,η_j^(|q+1|) 
   has full rank, the linear system 
   
      ℓ̅_i_1(η)'=0, …, ℓ̅_i'_2(η)'=0
  
   in the variables η_1^(|q|), …,η_j^(|q|) has full rank
   and is equal to the subsystem of (<ref>)
   obtained by canceling those equations whose index correspond to a complementary root. 
   The properties of p̅_i(η) and q_i(η) imply that 
   q̅_i(η) satisfies (c).
   We apply now Gaussian elimination to (<ref>) and obtain 
   
      η_k_1 = ℓ̅_k_1(η) + p̅_k_1(η), … ,η_k_2 = ℓ̅_k_2(η) + p̅_k_2(η)
  
   where ℓ̅_k(η) and p̅_k(η) 
   are obtained by applying the same row operations to 
   -ℓ̅_i(η)' and - q̃_i(η)
   respectively. Since the above mentioned properties of the last differential polynomials 
   do not change under 
   row operations, they also hold for ℓ̅_k(η) and p̅_k(η).
  
  
  
  
  We continue with the example for SL_4. The equations corresponding to non-complementary roots of height -1 are 
  
      h_1(η_6) = η_1' - η_4  + η_1^2 =0,     
      h_2(η_6) = η_2' + η_4 - η_5 + η_2(η_2-η_1 ) =0.
  
  Solving for their linear parts yields 
  
      η_4 = η_1'   + η_1^2,     
      - η_4 + η_5 = η_2'  + η_2( η_2-η_1)
  
  and if we add the first to the second equation we obtain 
  
      η_4 = η_1'   +  p̅_4 ,   η_5 = η_2' + η_1' + p̅_5
  
  where 
  
      p̅_4(η) = η_1^2 ,   p̅_5(η) = η_1^2 + η_2(η_2-η_1).
  
  One easily checks that the linear and non-linear parts in (<ref>) 
  satisfy respectively the statements of Lemma <ref> and <ref>.
  Furthermore the linear system 
  
      ℓ̅_4(η) = η_1' =0 ,  ℓ̅_5(η)=η_2' + η_1' = 0
  
  fulfills the first part of <ref>.
  For the non-complementary root of height -2 we consider the equation 
  
      h_4(η_6) = η_4' - η_6 - ( η_2 
      ( η_2 - η_1)) η_1 - η_3η_4    + η_5 η_1  - η_2' η_1 =0
  
  which we solve for its linear part. We obtain 
  
      η_6 = η_4'  - ( η_2 
      ( η_2 - η_1)) η_1 - η_3η_4    + η_5 η_1  - η_2' η_1 .
  
  We use the expressions in (<ref>) for the substitution of 
  η_4 and η_5 and obtain 
  
      η_6 =  η_1” + p̅_6(η)
  
  where 
  
      p̅_6(η) = 3 η_1  η_1'  +  η_1^3 -  η_3  η_1' 
       - η_3  η_1^2  .
  
  One sees that the linear and non-linear part satisfy <ref> and <ref> 
  respectively. The linear system
  ℓ̅_6(η)= η_1”=0 clearly satisfies the first part of <ref> and canceling the second equation of the system 
  
      ℓ̅_4(η)=0,  ℓ̅_5(η) = 0 ,
  
  which corresponds to a complementary root, yields the second part of <ref>. 
  The unique root of height -3 is a complementary root and so there is no equation to solve.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  We keep the notation of Lemma <ref>. Using the equivalent 
   system (<ref>) the differential polynomials 
   h_j_1(η_m), … , h_j_l(η_m) whose indices correspond 
   to the complementary roots reduce to differential polynomials 
  
      h_j_1(η)= ℓ̂_j_1(η) + p̂_j_1(η),  …,  
       h_j_l(η)= ℓ̂_j_l(η) + p̂_j_l(η)
  
  where ℓ̂_j_i(η) and p̂_j_i(η) have the following properties:
   
      
    *  The differential polynomial ℓ̂_j_i(η) ∈ C{η} is homogeneous of degree one and of order | r(j_i) |.
      
    *  The matrix formed by the coefficients of the linear system 
      
      ℓ̂_j_1(η)=0,  … ,  ℓ̂_j_l(η)=0
  
      in the variables η_1,…, η_l obtained by ignoring the derivatives 
      is quadratic and has full rank.
      
    *  The derivatives appearing in p̂_j_i(η) ∈ C{η} are at most of order |r(j_i)+1| and each term is of 
      degree greater than one.
  
  
  
  
  We fix the following notation. For a height q=-1,…, r(m) let j_r_1 and j_r_2 be the minimal and 
  maximal index among the indices of the complementary roots { j_1, … , j_l } such that 
  q ≤ r(j_r_1)=r(j_r_2) and let i_1 and k_1 be minimal and 
  i_2 and k_2 be maximal such that r(i_1)=q=r(i_2) and r(k_1)=q-1=r(k_2). Furthermore in the 
  following the index j (resp. i and k) runs between 
  j_r_1≤ j ≤ j_r_2 (resp. i_1≤ i ≤ i_2 and k_1 ≤ k ≤ k_2). If not otherwise stated 
  we ignore the derivatives of the variables when we consider a linear system.
  
  
  We prove by induction on the height q=-1,…, r(m) that using the equations 
  
      η_l+1 = ℓ̅_l+1(η) + p̅_l+1(η),    … ,  η_m = ℓ̅_m(η) + p̅_m(η)
  
  of Lemma <ref> the differential polynomials 
  h_j(η_m) reduce to  
  
      h_j_1(η)= ℓ̂_j_1(η) + p̂_j_1(η),  …, 
       h_j_2(η)= ℓ̂_j_2(η) + p̂_j_2(η)
  
  where ℓ̂_j(η) satisfy (a), p̂_j(η) fulfill (c) and the linear system 
  
      ℓ̅_k_1(η)=0,  …,  ℓ̅_k_2(η)=0, ℓ̂_j_1(η)=0,  …,  ℓ̂_j_r_2(η)=0
  
  has full rank.
  
  Let q=-1. In this case j_r_1=j_1 and k_1=l+1. We have the differential polynomials 
  
      h_j_1(η_m)=  
         η_j_1'+ ℓ_j_1(η_m)  +  q_j_1(η),  …,  
        h_j_2(η_m)=  
         η_j_2'+ ℓ_j_2(η_m)  +  q_j_2(η)
  
  where according to Lemma <ref> the element q_j(η) lies in 
  C[η_1,…, η_l] and each of its terms 
  is of degree greater than one and ℓ_j(η_m) is 
  homogeneous of degree one in the variables η_l+1,… ,η_k_2.
  From Lemma <ref> we obtain the equations    
  
      η_l+1 = ℓ̅_l+1(η) + p̅_l+1(η) ,
       …, 
      η_k_2 = ℓ̅_k_2(η) + p̅_k_2(η)
  
  where 
  
      ℓ̅_l+1(η)=0,…,ℓ̅_k_2(η)=0
  
  is a linear system in the variables 
  η_1', … , η_j_1 -1' of full rank and each term of 
  p̅_k(η) which lies in C[η_1,…,η_l ] 
  is of degree greater than one.
  We substitute the expressions of (<ref>) into ℓ_j(η_m) and 
  q_j(η) of the right hand sides of (<ref>) and obtain 
  
      h_j(η)=  η_j'+ ℓ_j(ℓ̅_l+1(η), …, 
         ℓ̅_k_2(η)) + 
         ℓ_j(p̅_l+1(η) , …, p̅_k_2(η) ) +  q_j(η)
  
  where j_1 ≤ j ≤ j_r_2. Since the variables η_j_1',…, η_j_r_2' do not appear among  
  the variables η_1', … , η_j_1 -1', we conclude that the linear system 
  
      ℓ̅_l+1(η)=0, …, ℓ̅_k_2(η)=0,   ℓ̂_j(η) = η_j'+ ℓ_j(ℓ̅_l+1(η),…,ℓ̅_k_2(η))=0
  
  with j_1 ≤ j ≤ j_r_2 has full rank and all variables 
  in ℓ̂_j(η) have order one. 
  Since ℓ_j(η_m) is linear in the variables η_k_1,…, η_k_2 and 
  p̅_k(η) as well as q_j(η) are polynomials where each 
  term is of degree greater than one, the same holds for
  
      p̂_j(η)= ℓ_j(p̅_l+1(η), …, p̅_k_2 (η)) +  q_j(η)
  
  with j_1 ≤ j ≤ j_r_2.
  
  
  Let q ≤ -2. We distinguish between the cases when there are complementary roots of 
  height q and when there are not. 
  In the first case we have the equations  
  
      h_j(η_m)=  η_j'+ ℓ_j(η_m) + 
         q_j(η_m)
  
  with j_1 ≤ j ≤ j_r_2
  where the non-linear parts q_j(η_m) are as in Lemma <ref> 
  and the linear parts form the linear system
  
      ℓ_j_r_1(η_m)=0, … , ℓ_j_r_2(η_m)=0
  
  of full rank in the variables η_k_1,…, η_k_2.  
  From Lemma <ref> we obtain the two systems of equations   
  η_i = ℓ̅_i(η) + p̅_i(η) and     
  η_k = ℓ̅_k(η) + p̅_k(η) 
  
  
  
  
  
  with i_1 ≤ i ≤ i_2 and k_1 ≤ k ≤ k_2.
  The corresponding linear systems 
  
      ℓ̅_i_1(η)=0, …, ℓ̅_i_2(η)=0   and  ℓ̅_k_1(η)=0, …, ℓ̅_k_2(η)=0
  
  in the variables η_1^(|q+1|), …,η_j'^(|q+1|) and 
  η_1^(|q|), …,η_j'^(|q|) respectively have full rank where j' is as j in Lemma <ref>. 
  We substitute the corresponding expressions among η_i_1,…, η_i_2 into 
  η_j_r_1',… , η_j_r_2' and the expressions for η_k_1,…,η_k_2 into 
  ℓ_j(η_m) of the right hand sides of the equations in   (<ref>). 
  We obtain  
  
      h_j(η_m) =  ℓ̂_j(η) + p̅_j(η)'+ ℓ_j(p̅_k_1(η) , …, p̅_k_2(η) ) +  q_j(η_m)
  
  with j_r_1≤ j ≤ j_r_2 where the homogeneous linear parts 
  
      ℓ̂_j(η) = ℓ̅_j(η)' + ℓ_j(ℓ̅_k_1(η),…,ℓ̅_k_2(η))
  
  are in variables η_1^(|q|), … , η_l^(|q|).
  By induction assumption the linear system
  
      ℓ̅_i_1(η)=0,  …,  ℓ̅_i_2(η)=0,  ℓ̂_j_1(η)=0, …,  ℓ̂_j_r_1-1(η)=0
  
  has full rank and by Lemma <ref> the linear 
  system 
  
      ℓ̅_k_1(η)=0, …, ℓ̅_k_2(η)=0
  
  is equivalent to the subsystem 
  
      ℓ̅_i_1(η)=0, …,  ℓ̅_j_r_1-1(η)=0
  
  of the system
  
      ℓ̅_i_1(η)=0, …, ℓ̅_i_2(η)=0
  
  
  
  
  
  obtained by canceling the
  rows which correspond to complementary roots of height q.
  We conclude that the linear system 
  
      ℓ̅_k_1(η)=0, …, ℓ̅_k_2(η)=0, 
      ℓ̂_j_1(η)=0,…, ℓ̂_j_r_1 -1(η)=0,  ℓ̂_j_r_1(η) = 0,  …, 
      ℓ̂_j_r_2(η) = 0
  
  has full rank. It follows from the order of the variables in ℓ̅_i(η) 
  and ℓ̅_k(η) that the variables in 
  ℓ̂_j(η) have order |q|. 
  This proves (a) and (b).
  
  We substitute the expression for η_l+1,…,η_i_2 which we obtain from 
  Lemma <ref> into the non-linear parts 
  
      p̂_j= p̅_j(η)' + ℓ_j(p̅_k_1(η) , …, p̅_k_2(η) ) +  q_j(η_m)
  
  with j_r_1≤ j ≤ j_r_2.
  The elements q_j(η_m) lie  in 
  C{η_1,…, η_i_1-1}[η_i_1  ,… ,η_i_2] according to Lemma <ref> and each 
  term of q_j(η_m) is of degree greater than one and the derivatives 
  appearing are at most of order one.
  Since the derivatives appearing in the expressions of η_i_1,…, η_i_2 and of 
  η_1,…, η_i_1-1 are at most of order |q+1| and |q+2| respectively, the derivatives 
  appearing in q_j(η_m) are at most of order |q+1|. The derivatives appearing in 
  p̅_k(η) and p̅_j(η) are at most of 
  order |q+1| and |q+2| respectively. Since ℓ_j(η_m) are homogeneous 
  of degree one in the variables η_k_1,…,η_k_2, we conclude that the derivatives appearing 
  in p̂_j(η) are at most of order |q+1|. 
  
  Assume that we are now in the case when there are no complementary roots of height q.
  By induction assumption we have that the linear system
  
      ℓ̅_i_1(η)=0, …, ℓ̅_i_2(η)=0,  ℓ̂_j_1(η)=0,…, ℓ̂_j_r_2(η)=0
  
  has full rank. Since the two systems 
  
      ℓ̅_i_1(η)=0, …, ℓ̅_i_2(η)=0   and  ℓ̅_k_1(η)=0, …, ℓ̅_k_2(η)=0
  
  are equivalent by Lemma <ref>, it follows that 
  
      ℓ̅_k_1(η)=0,  …,  ℓ̅_k_2(η)=0,  ℓ̂_j_1(η)=0,  …,  ℓ̂_j_r_2(η)=0
  
  also has full rank.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  We continue with the example for SL_4. 
  From Example <ref> we obtain that
  the coefficients in the linear representation of 
  ℓδ (Y) 
  of the basis elements for the non-complementary roots are
  
      h_3(η_6) = η_3' +  η_5 + q_3(η_6), h_5(η_6) = η_5' + η_6 + q_5(η_6), h_6(η_6) = η_6' +  q_6(η_6)  .
  
  Example <ref> provides the expressions
  
      η_4 = η_1'   +  p̅_4(η) ,  η_5 = η_2' + η_1' + p̅_5(η),  η_6 =  η_1” + p̅_6(η) .
  
  We substitute these accordingly into the differential polynomials of   (<ref>) 
  and obtain 
  
      h_3(η) =  η_3' +  η_2' + η_1' + p̂_3(η) , h_5(η) =  η_2” + 2 η_1” + p̂_5(η) ,  
      h_6(η) =  η_1”' +p̂_6(η)
  
  where 
  
      p̂_3(η) = p̅_5(η) + q_3(η),  p̂_5(η)= p̅_5(η)' + p̅_6(η) + q_5(η) ,  p̂_6(η)=p̅_6(η)' +  q_6(η).
  
  When we consider in the following a linear systems, we ignore the derivatives of the 
  variables, that is we understand the systems to be in the respective variables 
  η. We follow the argumentation of the proof of 
  Lemma <ref> to show that the linear system 
  
      ℓ̂_3(η)=η_3' + η_2' + η_1'=0 , ℓ̂_5(η) = η_2”+2η_1”=0,  ℓ̂_6(η)= η_1”'=0
  
  has full rank. 
  The linear system 
  
      ℓ̅_4(η) =η_1'=0,  ℓ̅_5(η)= η_2' + η_1'=0
  
  which corresponds to the roots of 
  height -2 has full rank. It follows that the linear system 
  
      ℓ̅_4(η)=0,  ℓ̅_5(η) =0,  ℓ̂_3(η)=η_3' + η_2' + η_1'=0
  
  has full rank, since the variable η_3' only 
  appears in ℓ̂_3(η).
  For the root of height -3 the linear system ℓ̅_6(η)= η_1”=0
  
  
  
  is equivalent to the subsystem ℓ̅_4(η) =0 of 
  
      ℓ̅_4 (η)=0,  ℓ̅_5(η)=0.
  
  It follows that the linear system 
  
      ℓ̅_6(η)=0 , ℓ̂_5(η) = η_2”+2η_1”=0,  ℓ̂_3(η)=η_3' + η_2' + η_1'=0
  
  has full rank, since ℓ̂_5(η) =0
  is obtained from adding ℓ̅_6(η)=0 to ℓ̅_5(η)'=0.
  Because the linear system ℓ̂_6(η)= η_1”'=0
  
  
  
  is equal to the linear system ℓ̅_6(η)'=0 we obtain that the linear system
  
      ℓ̂_6(η)= η_1”'=0,  ℓ̂_5(η)=η_2”+2η_1”=0 ,  ℓ̂_3(η)=η_3' + η_2' + η_1'=0
  
  has full rank.
  
  It is left to check that p̂_j(η) fulfills   <ref> for j=3,5,6.  
  Since all terms in p̅_5(η), p̅_6(η), q_3(η), q_5(η) and q_6(η) are of degree 
  greater than one, the same holds for p̂_j(η). We determine the maximal order of the 
  derivatives appearing in p̂_j(η). Since p̅_5(η) and q_3(η) are polynomials, so 
  is p̂_3(η). The derivatives appearing in p̅_6(η) and q_5(η) are at most of order one and so the
  same holds for p̂_5(η). Finally the derivatives appearing in q_6(η) are at most of order one and therefore the derivatives appearing in p̂_6(η) are at most of order two.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Let E=C⟨η⟩ (z,y) 
  and the elements h=(h_j_1(η), … , h_j_l(η))
  be as in Lemma <ref>  
  and <ref>. 
  Then E 
  is a Picard-Vessiot extension of C⟨h⟩ for 
  
      A_G (h ) = A_0^+ + ∑_k=1^l h_j_k(η)  X_j_k
   
  with differential Galois group G(C). The differential field  
  C⟨h⟩ is a purely differential transcendental extension of C of degree l. 
  
  
   
    We prove that the differential field extension 
   C ⟨h⟩ of C
   is a purely differential transcendental extension of degree l. 
   Clearly C ⟨η⟩ is a purely differential transcendental extension of C
   of degree l and we have the tower of differential fields 
   
      C ⊂ C ⟨h⟩⊂ C ⟨η⟩.
  
   From Lemma <ref> we obtain that 
  
      h_j_1(η)= ℓ̂_j_1(η) + p̂_j_1(η)
       , …,  
       h_j_l(η)= ℓ̂_j_l(η) + p̂_j_l(η).
  
  Moreover the properties of 
  ℓ̂_j_1(η), …, ℓ̂_j_l(η) and 
  p̂_j_1(η), …, p̂_j_l(η) 
  imply that after prolonging 
  h_j_1(η),…, h_j_l(η) to the order |r(j_l)|=|r(m)|
  the prolongations of ℓ̂_j_1(η),…,ℓ̂_j_l(η) 
  form a linear system in η_1^|r(m)| ,…, η_l^|r(m)| of full rank 
  and the derivatives appearing in the terms 
  in the prolongations of the non-linear parts 
  p̂_j_1(η),…, p̂_j_l(η) 
  are at most of order |r(m)+1|.
  We conclude that the transcendence degree of C ⟨η⟩ over
   C ⟨h⟩ is finite. 
   It follows that its differential transcendence degree is zero and so the differential transcendence
   degree of C ⟨h⟩ 
   over C is l by <cit.>. Since the extension is differentially
   generated by l elements, it is a purely differential transcendental extension.
   
   We show that 
   E 
   is Picard-Vessiot extension of C ⟨h⟩ and that its differential Galois group is G(C).
   Since E is a Picard-Vessiot extension
   of C ⟨η⟩ by 
   Proposition <ref>, its field of constants is C. 
   Moreover, by construction   
   
      Y= u((η_1, …,η_l,f_l+1(η),
        …, f_m(η)) )
        n(w̅) 
       t(z) u(y)
  
   is a fundamental solution matrix where f_l+1(η),… ,f_m(η) are as 
   in Lemma <ref>. We need to show that the 
   entries of Y generate E as field over 
   C ⟨h⟩. Clearly E
   contains C ⟨h⟩ (Y_ij) and we need to 
   prove the other inclusion. Since Y lies in G we can compute 
   its Bruhat decomposition over C ⟨h⟩ (Y_ij). Using the uniqueness of the elements of the Bruhat decomposition we get that η, f_l+1(η), …, f_m(η), 
   z and y can be represented as rational expressions in the entries of Y. Since C ⟨h⟩ (Y_ij)
   is a differential field it follows that it is equal to E and so E is generated as a field by the entries of Y over C ⟨h⟩.
   We conclude that E is a Picard-Vessiot extension of C ⟨h) ⟩.
   We proved above that h are differentially algebraically independent over C and so it follows from Proposition 5.3, Lemma 6.8 and Theorem 4.3 of 
   <cit.> applied to A_G(h) that the differential Galois group is G(C).
   
  
  
  
  Theorem <ref> shows that for the differential field 
  E=C⟨η⟩( z,y)
  and the group G(C) a construction as explained in the 
  introduction is possible. Let 
   
      Y =u( (η, f_l+1(η)
        , …, f_m(η)) )  n(w̅) 
       t (z) u(y)
  
  and for g ∈ G(C) consider the Bruhat decomposition of Y g.
  Let (x̅,z̅,y̅)
  be the corresponding normal form coefficients. Then the map 
  
      φ_g: E  →  E ,  
       ((η_1, …, η_l, f_l+1(η)
        , …, f_m(η)) , z,y) ↦ 
       (x̅,z̅,y̅)
  
   is a differential C-automorphism and the group G(C) acts on E by 
  
      G(C) × E → E,  (g,x) ↦φ_g(x)
  
  where the field of invariants is E^G = C⟨h⟩. 
  Indeed, it follows from Theorem <ref> that for g ∈ G the map φ_g is actually the Galois automorphism induced by Yg of the 
  Picard-Vessiot extension E of C ⟨h⟩, 
  since the normal form coefficients of the Bruhat decompositions of Y and Yg can be 
  written as rational expression in the entries of Y and Y g respectively.
  
  
  
  
  
  
  
  
   
  
  
  
  
  In this example we consider the group G and its Lie algebra 𝔤 of type G_2. We compute the general extension field E with its Liouvillian solutions and the invariants generating the fixed field. 
  Let Δ = {α̅_1, α̅_2 } be simple roots and 
  we number the six negative roots according to Chapter <ref> as 
  
      β_1= -α̅_1,  β_2= -α̅_2,  β_3= -α̅_1 - α̅_2,  β_4= -2α̅_1 - α̅_2,  β_5= -3α̅_1 - α̅_2 
      and  β_6= -3α̅_1 - 2 α̅_2.
  
  The complementary roots are {β_2, β_6 }. We take the 
  representation of 𝔤 as in <cit.>.
  From the Chevalley basis presented there we compute the root group 
  elements 
  u_1(η_1), …, u_6(η_6) belonging to the negative roots for 
  indeterminates η_6=(η_1, …, η_6) and the torus elements  
  t_1(z_1), t_2(z_2) for indeterminates z=(z_1, z_2). 
  For the simple reflection w_α̅_1 and w_α̅_2
  we choose the representatives 
  
      n(w_α̅_1)   =    -E_11+E_72-E_63-E_54+E_45-E_36-E_27
      
      n(w_α̅_2)   =      E_11+ E_22 -E_34+E_43+E_55-E_67+E_76
  
  and obtain the representative n(w̅)=(n(w_α̅_2)n(w_α̅_1))^3 for the 
  longest Weyl group element w̅. With further indeterminates 
  y=(y_1, …, y_6) we define 
  
      Y=u_1(η_1) ⋯ u_6(η_6) n(w̅)t_1(z_1)t_2(z_2)u_1(y_1) ⋯ u_6(y_6)
  
  and compute its logarithmic derivative
  
      ℓδ (Y)= ∑_i=1^6 a_iX_i +d_1 H_1+d_2 H_2 +
      ∑_i=1^6 a_-β_iX_-β_i
   
  where the coefficients are differential 
  rational expressions in the indeterminates  
  η_6, z and y.
  In <cit.> the parameter differential equation with Galois group G is defined by the matrix 
  A_0^+ + t_1 X_2 + t_2 X_6 with parameters t_1, t_2 and so  
  we determine in the following η_6, z and y such that the logarithmic derivative of Y satisfies  
  
      ℓδ (Y) = A_0^+ + h_2(η) X_2+ h_6(η) X_6
  
  where η=(η_1, η_2). In a first step we require that 
  
      a_-β_6=0, …, a_-β_3=0,  a_-β_2=1,  a_-β_1=1,   d_1=0,  d_2 =0.
  
  It is possible to solve these equations successively for y_6', …, y_1' and z_1'/z_1,z_2'/z_2. Indeed, one can check that 
  in each step we can solve the equation for the first derivative or the logarithmic derivative of the corresponding variable and 
  substitute the obtained expression in the 
  remaining equations. This process yields the solutions 
   
      z_1= e^∫η_1  ,  z_2=e^∫ - η_2   , y_1=∫-1/z_1^2 z_2, y_2 =∫ - z_1^3 z_2^2,  
      y_3 = ∫ y_1 y_2', 
      
      y_4= ∫ -3 y_1 y_5'-3 y_1^2 y_3'+ y_1^3 y_2',  
      y_5= ∫ -2 y_1 y_3' + y_1^2 y_2' ,  
      y_6 = ∫ -3 y_3 y_5' - y_2 y_4'  .
  
  The differential field extension E=C⟨η⟩(z,y)
  of C⟨η⟩ is a Picard-Vessiot extensions with 
  differential Galois group B^- according to Proposition <ref> and it is our general extension field.
  It is left to determine the two generators of the field of invariants.
  If we substitute these solutions into the logarithmic derivative of Y we obtain   
   
      lδ(Y)=A_0^+ + h_1(η_6) X_1+ … + h_6(η_6) X_6
   
  with 
   
      h_ 1(η_6)    =   η_1'-η_3+η_1^2  ,  
       
      h_ 2(η_6)    =   η_2' +3 η_3+η_2^2-3 η_1 η_2    
      
      h_3(η_6)     =   η_3'- h_2(η_6) η_1 -2 η_5 - η_1 η_3 , 
      
      h_ 4(η_6)    =   η_5'+ η_1 η_5 -η_4 + η_3^2 +  h_2(η_6) η_1^2  
      
       h_5(η_6)    =   η_4' - h_ 2(η_6) η_1^3 - η_6 + η_4 (3 η_1 - η_2)
      
       h_6(η_6)    =   η_6' + h_2(η_6) (η_1^3η_2 - 3η_1^2 η_3) + 
      η_2 η_6 + 3 η_5^2 - 2 η_3^3  .
  
  Next we require that the coefficients of basis elements corresponding to the non-complementary roots vanish. This leads to the system of equations 
  h_1(η_6)=0, h_3(η_6)=0, h_4(η_6)=0 and h_5(η_6)=0.  
  These equations depend linearly on the 
  variables η_3, η_5, η_4 and η_6. 
  Successively we solve the 
  equations for the corresponding variables and substitute the obtained 
  expressions for the variables into the remaining equations. This process yields the equations 
  
      η_3= f_3(η),  η_5= f_5(η),  η_4= f_4(η),  η_6= f_6(η)
  
  and the invariants
   
      h_1(η)    =   η_2' +3 (η_1' + η_1^2) +η_2^2-3 η_1 η_2 
      
       h_6(η)    =    -1/4((2 η_1”'+4 η_1 η_1”+6 η_1'^2+(4 η_1^2-2 h_1) η_1'-2 η_1 h_1'+2 η_1^4+2 h_1 η_1^2) η_2'
            -2 η_1^(5)-10 η_1 η_1^(4)+(-26 η_1'+2 η_2^2-6 η_1 η_2-16 η_1^2+2 h_1) η_1”'-19 η_1”^2
            +(6 h_1'-90 η_1 η_1'+4 η_1 η_2^2-12 η_1^2 η_2-14 η_1^3+8 h_1 η_1) η_1”-18 η_1'^3 +(6 η_2^2
            -18 η_1 η_2-39 η_1^2+2 h_1) η_1'^2+ (6 h_1”+10 η_1 h_1'+12 η_1^2 h_1+(4 η_1^2-2 h_1) η_2^2
            +(6 h_1 η_1-12 η_1^3) η_2) η_1'+2 η_1 h_1”'+4 η_1^2 h_1”+(-2 η_1 η_2^2+6 η_1^2 η_2-2 η_1^3) h_1' 
             +(2 η_1^4+2 h_1 η_1^2) η_2^2+(-6 η_1^5-6 h_1 η_1^3) η_2+5 η_1^6 + 6 h_1 η_1^4-3 h_1^2 η_1^2)
   
  where we write shortly h_1 for h_1(η). 
   
  
   
   
  
  
  § FULL G-PRIMITIVE PICARD-VESSIOT EXTENSIONS AND 
  THE GENERAL EXTENSION FIELD
   
   
  Let E be a full G-primitive Picard-Vessiot extension of a differential field F with defining matrix A ∈𝔤(F). In this 
  chapter we pursue the question whether there exist a specialization of the parameters
  
      h=(h_j_1,…,h_j_l) ↦f=(f_1,…,f_l)
  
  to elements of F such 
  that E is a Picard-Vessiot extension of F for the specialized matrix  A_G(f).
  If A is gauge equivalent to some A_G(f) by an element of G(F), 
  then the corresponding specialization has trivially the desired properties. 
  We develop in this 
  chapter a criterion 
  depending on the differential subfield F(x) of E, where x 
  are the first 
  m normal form coefficients of a fundamental matrix for A, to decide whether the orbit of 
  A under gauge transformation intersects the set of matrices of shape A_G(f).
  
  The orbit of a matrix in 𝔤(F) under G(F) depends on the differential field F. 
  In the special case when F is equal to the field constants C, the gauge transformation 
  reduces to the adjoint action. The orbit structure in the algebraic case was studied by B. Kostant 
  in <cit.>. According to <cit.> 
  the orbits of the matrices of shape A_G(f) are in 
  bijective correspondence to the orbits of maximal dimension in 𝔤(F). 
  Moreover in the special 
  case when F is differentially closed the logarithmic derivative from G(F) to 
  𝔤(F) is surjective and so any two matrices are gauge equivalent, that is there is 
  only a single orbit. In both special cases we have a description of all matrices which are gauge 
  equivalent to matrices of shape A_G(f) and they show how the orbit structure 
  varies with the differential field F.
   
  For an algebraically closed differential field F the following lemma shows that matrices of 
  a specific subset of 𝔤(F) are gauge equivalent to matrices of shape A_G(f).
  
   Let F= F and A ∈ A_0^+(s) +  𝔟^-(F) with nonzero s=(s_1,…,s_l)
   in F. Then A is gauge equivalent by an element of B^-(F) to a matrix of shape 
   A_G(f).  
  
  For proof of Lemma <ref> see <cit.> 
  
  Let A be an element of an orbit of a matrix of shape A_G(f).
  Then there are w ∈𝒲, u ∈ (U^-)'_w and 
  nonzero s=(s_1,… ,s_l) of F such that A is an element of the plane
  
      Ad(u n(w)) (A_0^+(s) + 𝔟^-)  + ℓδ (u)
  
  where n(w) is a representative for w and (U^-)'_w is as in Theorem <ref> 
  with the roles of the positive and negative roots interchanged. 
  
  
  Since A is an element of the orbit of A_G(f), there is g ∈ G(F) such that 
  
      A= Ad(g)(A_G(f) ) + ℓδ (g).
  
  By Theorem <ref>, where we interchange the role of the positive and negative roots, there exists w ∈𝒲, u ∈ (U^-)'_w(F), t ∈ T(F) 
  and u̅∈ U^-(F) such that g=u n(w) t u̅ with n(w) a 
  representative of w in the normalizer of T. It follows that  
  
      A = Ad( u n(w)) ( Ad(t u̅)( A_G(f)) + ℓδ (t u̅) 
      ) + ℓδ (u n(w)) .
  
  Finally Remark <ref> and Proposition <ref> imply 
  that the gauge-transform of A_G(f) with t u̅ lies in the plane 
  A_0^+(s) + 𝔟^-(F) for some nonzero s=(s_1,… ,s_l)
  of F.
  
  
  
  
   Let w ∈ W and fix a representative n(w) in the normalizer of T. Moreover let  
   C(x, z,y) be the rational function field in the indeterminates x=(x_1,… , x_m), z=(z_1,…,z_l) and 
   y=(y_1,…,y_m).
   Then there are x̅=(x̅_1,… , x̅_m) in C(x) 
   and v̅=(v̅_1,… , v̅_m) and nonzero 
   z̅=(z̅_1,… , z̅_l) in C(x,z)
   such that 
   
      n(w) ·( u(x) n(w̅) t(z)u(y) )
       = u(x̅) n(w̅) t(z̅)
       u(v̅) u(y).
  
  
  
  
   For a simple root α̅_i let 
   G_i be the centralizer of kern(α̅_i) ⊆ T in G. Then 
   G_i is of semisimple rank 1 and it is generated according to 
   <cit.> by the subgroups 
   U_α̅_i, U_-α̅_i and T. 
   We prove that a specific relation holds in G_i. Let 
   
      u̅_α(x)=[ 1 x; 0 1 ], u̅_- α(x)=[ 1 0; x 1 ], t̅(x)=[   x   0;   0 1/x ] , n̅ =[  0  1; -1  0 ]
  
   be the standard representation of SL_2(C) where α is the unique simple root and 
   n̅ a representative of the unique nontrivial Weyl group element.
   Then a simple computation shows that
    
      n̅u̅_-α(x) =  
        u̅_- α(-1/x) t̅(x) u̅_α(1/x).
  
   Applying the injective group homomorphism from SL_2(C) to G_i which is 
   defined by 
   
      u̅_α(x) ↦ u_α̅_i(x),  u̅_-α(x)  ↦ u_-α̅_i(x),  t̅(x) ↦ t_i(x) ,  n̅↦ n(w_α̅_i)
   
   to the last equation we obtain that the corresponding relation is also true in G_i.
   
   For a simple root α̅_i  let = Φ^- ∖{ - α̅_i }. 
   By <cit.> the root groups U_β_j with β_j ∈ 
   directly span a subgroup U_ which is normalized by G_i and satisfies
   U^-= U_-α̅_i U_ = U_  U_-α̅_i. We conclude that 
   
      n(w_α̅_i) u(x) = 
       n(w_α̅_i) u_(x̅_) 
       u_-α̅_i(x̅_i) =
       u_(x̅_) 
       n(w_α̅_i) u_-α̅_i(x̅_i)
  
   where x̅_i and x̅_=(x̅_1,…, x̅_i-1, x̅_i+1,…, x̅_m) are elements in C(x) and 
   
      u_(x̅_) =u_1(x̅_i) ⋯ u_i-1(x̅_i-1)u_i+1(x̅_i+1)⋯ u_m(x̅_m) .
  
  Since n(w_α̅_i) u_-α̅_i(x̅_i) ∈ G_i we obtain from 
  the above relation that  
  
      u_(x̅_) 
       n(w_α̅_i) u_-α̅_i(x̅_i) = 
       u_(x̅_) 
       u_-α̅_i(-1/x̅_i) t_i(x̅_i)u_α̅_i(1/x̅_i) = 
      u( x̅) 
       t_i(x̅_i)u_α̅_i(1/x̅_i)
   
  with suitable x̅ in C(x). The torus is normalized by 
   n(w̅) and the adjoint action of n(w̅) maps the positive root groups 
   bijectively to the negative root groups. So we have  
   
      u(x̅) 
       t_i(x̅_i)u_α̅_i(1/x̅_i)
       n(w̅) t(z)u(y) = 
       u(x̅) n(w̅)
       t(ẑ) u(v̂)
        t(z)u(y)
  
  with nonzero ẑ and v̂ in C(x). 
  Finally, since the torus normalizes U^-, we obtain 
    
      u(x̅) n(w̅)
       t(ẑ) u(v̂)
        t(z)u(y)=
       u(x̅) n(w̅)
       t(z̅) 
       u(v̅) u(y)
  
   with nonzero z̅ and v̅ in C(x,z) which completes the proof for w_α̅_i∈𝒲.
   
   Now let w ∈𝒲 be arbitrary. 
    By <cit.> it is the product of simple reflections 
   and so the statement of the lemma follows from applying successively the above result. 
   
  
  
  
  
  Let E/F be a full G-primitive Picard-Vessiot extension with matrix Y ∈ G(E) 
  and normal form decomposition 
  
      Y=u(x) n(w̅) t(z)u(y).
  
  Moreover let u ∈ U^-(F) and w ∈𝒲 with representative n(w). 
  Then there are x̅=(x̅_1,…, x̅_m) in F(x)
  and nonzero z̅=(z̅_1,…, z̅_l) and 
  v̅=(v̅_1,…,v̅_l) in F(x,z)
  such that
   
      ( n(w) u ) ·(  u(x) n(w̅) t(z)u(y) )
       =  u(x̅) n(w̅) t(z̅)
       u(v̅) u(y) .
  
  
  
   Since u and u(x) are elements of U^-(F(x)), 
   there are x̂=(x̂_1,…,x̂_m) in F(x)
   such that u ·u(x)=u(x̂).
   Then the statement follows from applying Lemma <ref> to
   
      n(w) u(x̂) n(w̅) t(z)u(y) .
   
  
  
  
  
  Let E be a full G-primitive Picard-Vessiot extension of F with normal 
  form decomposition
  
      Y=u(x) n(w̅) t(z)u(y).
  
  The fundamental theorem yields that E is a Picard-Vessiot extension of 
  E^U^- with differential Galois group U^-(C) and it follows from 
  Proposition <ref> that E^U^- is  
  F(x,z). Since 
  u(x) n(w̅) t(z) lies in 
  G(F(x,z)), the matrix u(y) is 
  a fundamental matrix for the extension E of F(x,z).
  Clearly the defining matrix ℓδ (u(y)) for the 
  extension lies in the 
  subalgebra 𝔲^-(F(x,z)). We call 
  u(y) the 
  unipotent part of the normal form for Y. 
  
  
  
  
  
   A gauge transformation of the logarithmic derivative of the unipotent part by an element u(v̅) ∈ U^-(F(x,z)) 
   as in Corollary <ref>
   is called an equivariant transformation of the unipotent part.
  
   
   
  
   
  
  Let E be a full G-primitive Picard-Vessiot extension of F with normal 
  form decomposition
  
      Y=u(x) n(w̅) t(z)u(y).
  
  Then ℓδ (Y) is gauge equivalent by an element of G(F) to a matrix 
  of shape A_G(f) if and only if there is an equivariant transformation 
  of the unipotent part to a principal nilpotent matrix in normal form A^-_0(s) with 
  s=(s_1,…,s_l) in F(x,z).
  
  
   
  First assume that there is g ∈ G(F) and f=(f_1,…,f_m) in F such that 
  
      ℓδ (Y)= Ad(g)( A_G(f) ) + ℓδ (g).
  
  We conclude with Theorem <ref> that there are w ∈𝒲, u ∈ U^- and 
  nonzero s̅=(s̅_1,…,s̅_l) in F such that 
  
      Ad(n(w) u) (ℓδ(Y)) + ℓδ (n(w) u)= A̅∈ A_0^+ (s̅) + 𝔟^-(F)
  
  where n(w) is a representative of w. 
  The last equation implies that n(w) u Y is a fundamental solution matrix for A̅. 
  Moreover from Corollary <ref> we obtain that 
  there are x̅=(x̅_1,…, x̅_m) in F(x)
  and nonzero z̅=(z̅_1,…, z̅_l) and 
  v̅=(v̅_1,…,v̅_l) in F(x,z)
  such that
  
      n(w)u  Y= (n(w) u) ·( u(x) n(w̅) t(z)u(y) )
      = u(x̅) n(w̅) t(z̅)
       u(v̅) u(y)
  
  and so u(v̅) is an equivariant 
  transformation of the unipotent part u(y). Further the 
  logarithmic derivative of the fundamental matrix n(w)u Y computes as 
  
      A̅ = ℓδ (u(x̅)) + Ad(u(x̅) n(w̅)) ( ℓδ (t(z̅)) ) + Ad(u(x̅) n(w̅) t(z̅) ) (ℓδ (u(v̅) u(y)) ).
  
  The first two summands lie in 𝔟^-. Since w̅ sends the negative roots 
  bijectively to the positive roots and in particular the negative simple roots to the simple roots, we 
  conclude that there are s=(s_1,…,s_l) in F(x,z) 
  such that 
  
      ℓδ (u(v̅) u(y))=A^-_0 (s).
  
  Hence the logarithmic derivative of 
  u(y) is gauge equivalent by
  u(v̅) to a principal nilpotent matrix in normal form.
  
  Next we assume that there is an equivariant transformation of the logarithmic 
  derivative of 
  u(y) to a matrix of shape A^-_0 (s)
  with nonzero s=(s_1,…,s_l) in F(x,z).
  This means that there are w ∈𝒲, u ∈ U^-(F), 
  x̅=(x̅_1,…, x̅_m) in F(x)
  and nonzero z̅=(z̅_1,…, z̅_l) and 
  v̅=(v̅_1,…,v̅_l) in F(x,z)
  such that
   
      ( n(w) u ) ·(  u(x) n(w̅) t(z)u(y) )
       =  u(x̅) n(w̅) t(z̅)
       u(v̅) u(y)
  
  and such that 
  
      ℓδ (u(v̅) u(y))=A^-_0 (s).
  
  The logarithmic derivative of the right hand side of the last equation 
  is
  
      ℓδ (u(x̅)) + Ad(u(x̅) n(w̅)) ( ℓδ (t(z̅)) ) + Ad(u(x̅) n(w̅) t(z̅) ) (ℓδ (u(v̅) u(y)) ).
  
  As above the first two summands lie in the subalgebra  𝔟^-.
  Since the logarithmic derivative of u(v̅)u(y) is
  A^-_0 (s), it is send by the adjoint action with n(w̅) t(z̅) to a matrix of shape A^+_0 (s̅) with 
  nonzero s̅=(s̅_1,…,s̅_l) in F(x,z).
  Further the adjoint action with u(x̅) maps A^+_0 (s̅)
  to the plane A^+_0 (s̅) + 𝔟^- and so combining all results 
  we get that the logarithmic derivative of n(w) uY lies in the plane  
   A_0^+(s̅) + 𝔟^-(F) and that s̅=(s̅_1,…, s̅_l) are actually in F. The statement then
   follows from applying Lemma <ref>.
  
  
  
   A full G-primitive Picard-Vessiot extension such that there is an 
   equivariant transformation of the unipotent part to a principal nilpotent matrix in normal form is 
   called a full G-primitive Picard-Vessiot extension with normalisable unipotent part.
  
  
  
  
  The Picard-Vessiot extension C ⟨η⟩ (z, y)
  of C⟨h⟩ and the defining matrix A_G(h) of 
  Theorem <ref> 
  are generic for every full G-primitive Picard-Vessiot extension L of an 
  algebraically closed differential field F with normalisable unipotent part
  in the following sense:
  
  
    * 
  There exists a specialization σ : C {h}→ F such 
  that L is a Picard-Vessiot extension of F for 
  A_G(σ(h)) and the defining matrix of L is 
  gauge equivalent to A_G(σ(h)). 
  
    *    
  There exists a specialization σ: C{η}[z,z^-1 ,y] → L such that 
  
      L=F(σ(Y))
  
  where Y is the fundamental matrix for A_G(h).  
  
    * 
  For every specialization σ : C {h}→ F the differential 
  Galois group of a Picard-Vessiot extension for A_G(σ(h)) is a subgroup of G(C).
  
  
  
  
  
  Let Y̅∈ G(L) be a fundamental solution matrix for the extension L of F. Since the extension has a normalisable unipotent part 
  and F is algebraically closed it follows from 
  Theorem <ref> 
  that there are g ∈ G(F) and f=(f_1,…,f_l) in F such that 
  ℓδ (Y̅) can be gauge transformed by g to A_G(f). 
  This means that L/F is a full G-primitive Picard-Vessiot extension for A_G(f) with 
   fundamental solution matrix gY̅∈ G(L). Furthermore we have a normal form decomposition
   
      g Y̅=u(x̅_m) n(w̅)t(z̅)
       u(y̅)
  
   where  x̅_m=(x̅_1,…,x̅_m), 
   z̅=(z̅_1,…,z̅_l) and 
   y̅=(y̅_1,…,y̅_m) 
   in L are the normal form coefficients. For the first 
   l entries x̅=(x̅_1,…, x̅_l) in x̅_m we define the specialization
   
      σ : C {η}→ L,  η↦x̅.
  
   Since the logarithmic derivative of gY̅ is A_G(f) it follows 
   from Theorem <ref> that 
   h_j_i(x̅)=f_i
   and so the restriction   
    
      σ|_C{h} : C {h}→ F,  h↦f
  
   satisfies statement <ref>.  
   Furthermore by the same argument we have that
    
      x̅_l+1 = f_l+1(x̅) , … ,  
        x̅_m = f_m(x̅)
  
   and that ℓδ(t(z̅)u(y̅))=A_L(x̅). Thus we can extend σ to a 
   specialization 
   
      σ : C {η}[z,z^-1,y] 
       → L ,   (η , z, y)    ↦ 
        (x̅, z̅, y̅)
  
   which satisfies  
    
      σ(Y)=u(σ(η,f_l+1(η), … ,
       f_m(η))) n(w̅) t(σ(z))
       u(σ(y)) = 
       u(x̅)n(w̅)  t( z̅) 
        u( y̅).
  
   This shows the second statement of the theorem. Finally the third statement 
   follows from <cit.>, since for every specialization 
   σ: C{h}→ F the matrix 
   A_G(σ(h)) lies in 𝔤. 
  
  
  
  
  
  
  The proof of Theorem <ref> shows that every full G-primitive 
  Picard-Vessiot extension with normalisable unipotent part is determined by 
  l=rank(G) many elements of L. More precisely, the elements x̅=(x̅_1,…,x̅_l) 
   differentially generate the intermediate extension L^B^-/F and, since L/F has normalisable unipotent 
   part, they determine the Liouvillian extension L/L^B^- by A_L(x̅).
  
  
  
  
  
  
  plain
  
  
  
  
  