Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$.

Let $X$ be the (complete) flag variety of $G$ dimension $n$, hence we can assume that $X=G/B$. Furthermore let $x \in X$ be the unique fixed point of $B$ under the natural left action of $G$ on $X$. Then denote for $w \in W$ by $X_w=Bwx$ the corresponding schubert cell in $X$ with dimension $l(w)$.

According to p. 51 in Brylinskis paper "Differential operators on the flag varieties" we have that the formal character of $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ is given by $\frac{e^{-w(\rho)-\rho}}{\prod_{\alpha \in R(B)}(1-e^{-\alpha})}$. Here $R(B)$ is the set the roots of $B$, hence the positive roots of $G$ with respect to $B$ and $\rho$ is the half sum of all positive roots of $G$ with respect to $B$. Finally he concludes that $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ has highest weight $-w(\rho)-\rho$.

He refers to Kempfs "The Grothendieck-Cousin Complex of an Induced Representation" for a proof of this result. But when I look it up there, concretely Lemma 12.8, I would rather say that the formal character of $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ is given by $\frac{e^{w(\rho)+\rho}}{\prod_{\alpha \in R(B)}(1-e^{\alpha})}$. So the sign changed everywhere. Hence it would have lowest weight $w(\rho)+\rho$.

Can anyone solve my confusion?

  • $\begingroup$ I am not familiar with Kempf's paper but the sign change suggest that there should be some kind of "reversion" -- switching positive roots with negative, one Borel with the opposite etc. In fact, Brylinski's filtration is ascending but Kempf's is descending (see Lemma 12.1). $\endgroup$ Jun 15 '20 at 11:06
  • $\begingroup$ That being said, as far as I can see, the results are really at odds. $\endgroup$ Jun 15 '20 at 11:06
  • $\begingroup$ That was my first thought too, but as far as I see they both started with some arbitrary Borel. That one chain is ascending and the other one descending follows just by considering in one case the dimension and in the other one the codimension. $\endgroup$
    – KKD
    Jun 15 '20 at 16:44
  • $\begingroup$ You are correct. I spend few hours with the Kempf's paper and as far as I can understand it is self-consistent. But I don't understand corollary 11.10 and how he uses it to get 12.8. $\endgroup$ Jun 16 '20 at 12:50
  • 1
    $\begingroup$ How I understand to get lemma 12.8: Let $C$ be a schubert cell with respect to a Borel $B$ containing a $T$-fixed point $g$. By proposition 6.4 we have for the open nbh $N(C)$ of $C$ a T-equivariant isomorphism $N(C) \cong Z \oplus U$, with $Z=\bigoplus_{\chi \in J(C)}V(\chi)$ and $U=\bigoplus_{\chi \in -K(C)}V(\chi)$. Also $C=Z \oplus 0$. Hence with $A$ and $B$ from corollary 11.10 (and the discussion before the corollary) being $-K(C)$ resp. $J(C)$, we have that $Z_n=C$. The rest is plugging in corollary 11.10. What is your problem with 11.10? $\endgroup$
    – KKD
    Jun 16 '20 at 14:53

The formula 12.8 in Kempf follows by plugging in 6.5 into 11.10 (see comment by CJS under the question). In the statement of 11.10 Kempf defines $[\chi]$ to denote the isomorphism class of $V(\chi).$ Now $V(\chi)$ is the dual of $\textbf{V}(\chi)$ (see page 313 or 380) and the torus $T$ acts on $\textbf{V}(\chi)$ by $(g,v) \mapsto \chi(g) v$ (see p. 313) which makes the weight/character of $V(\chi)$ to be equal to $-\chi.$ This reconciles Kempf's result/notation with Brylinski's statement.

  • $\begingroup$ But then my question is still, why the characteristics in Kempf‘s work p.313/314 are the same and have not opposite sign? $\endgroup$
    – KKD
    Jun 19 '20 at 16:55
  • $\begingroup$ My understanding is that he just defines the characteristics of algebraic representations in a way which is incompatible with how $T$ acts. I have no idea why. $\endgroup$ Jun 19 '20 at 21:49

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