# A trilinear forms question about representations of real linear group.

This might be a naive question. Suppose $\pi_i$ are irreducible generic unitary representation of $GL_{n}(\mathbb{R})$ with its associated Whittaker models $\mathcal{W}(\pi_i)$. Let $E_m$ be the space of homogeneous polynomials of degree $m$ on $\mathbb{R}^n$, then $GL_n(\mathbb{R})$ acts on this space via $(g.P)(x)=P(x.g)$, where $x.g$ denotes $g$ acts on $x$ from the right. Let $e=(0,...,0,1)\in \mathbb{R}^n$. Finally let $s_0$ be some complex number. Let $Z$ be the center of $GL_n$, $N$ the standard maximal unipotent subgroup.

Now suppose that there is a nontrivial continuous trilinear form $L:V_1\times V_2 \times E_m\rightarrow \mathbb{C}$ satisfying $L(g.W_1,g.W_2,g.P)=|detg|^{-s_0}L(W_1,W_2,P)$ for all $W_1,W_2,P$ . Then is it always true that $$\int_{ZN\GL_n}W_1(g)W_2(g)P(eg)|detg|^{s_0}dg < \infty ?$$

In other words, the above integral gives one realization of $L$.

-
If your integral as a function of $s_0$ has analytic continuation to $\mathbb C$ then you may define a linear form even if the integral doesn't converge. Did you have a look at Jacquet "Automorphic forms for GL(2), part II" (Springer LN) where he considers such integrals for $n=2$ ? – Paul Broussous Nov 17 '11 at 9:53
Sorry I meant "meromorphic continuation" – Paul Broussous Nov 17 '11 at 9:53
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9989275932312012, "perplexity": 144.6134567156874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111392.88/warc/CC-MAIN-20160428161511-00034-ip-10-239-7-51.ec2.internal.warc.gz"}