Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $W$ be the normalized spherical Whittaker function attached to a spherical representation $\pi$ on $GL_n(k)$, where $k$ is a $p$-adic field and $n\ge 3$.

I'm faced with the slightly-odd integral $$\int_{k^\times} \left|W\left(\matrix{y&&&\cr &\ddots&&\cr &&y&\cr &&&1}\right)\right|^2 |y|^s\ dy$$ where ${\rm Re}(s)$ is sufficiently large to get convergence.

Does anyone have any ideas about calculating this?

Recall the Casselman-Shalika-Shintani formula, $$W(\varpi^J)=\delta_B^{1/2}(\varpi^J){\rm Tr}\left(\rho_J\left(A_\pi\right)\right)$$ where $J=(j_1,\ldots,j_n)\in\mathbb Z^n$ satisfies $j_1\ge j_2\ge\ldots\ge j_n$, $\varpi^J$ is the diagonal matrix with $j$-th entry $\varpi^{j_i}$, $\delta_B$ is the modular character of the Borel subgroup, $\rho_J$ is the representation of $GL_n(\mathbb C)$ with highest weight $(j_1,\ldots,j_n)$, and $A_\pi$ is the matrix of Satake parameters of $\pi$. Using this, the integral can be written as $$\sum_{i\ge 0} q^{-i(s+n-1)}{\rm Tr}\left(\rho_{(i,\ldots,i,0)}(A_\pi)\otimes\rho_{(i,\ldots,i,0)}(\bar A_\pi)\right)$$ where the shift by $n-1$ comes from $\delta_B(\varpi^{(i,\ldots,i,0)})=\prod_{j=0}^{n-2}|\varpi^i|^{n-1-2j}$.

This is basically a sub-sum of the formula used to get the Rankin-Selberg $L$-function for $\pi\otimes\tilde\pi$, so it should have a reasonable answer.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.