# 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 $L$.

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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