[Paper Review] Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif $p$-adique - le cas des groupes classiques

Volker Heiermann|arXiv (Cornell University)|Apr 28, 2008

Advanced Algebra and Geometry11 references22 citations

TL;DR

This paper computes the endomorphism algebra of certain induced projective generators in the category of smooth representations of $p$-adic classical groups (symplectic, orthogonal, or inner forms of general linear groups). Using Bernstein's framework and analyzing intertwining operators, it proves the algebra is isomorphic to a semidirect product of a Hecke algebra with parameters and a finite group algebra, generalizing known results and providing explicit structure for these algebras.

ABSTRACT

For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators à la Bernstein of the category of smooth representations of $G$ and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected $p$-adic group.

Motivation & Objective

• To determine the structure of the endomorphism algebra of induced projective generators in the category of smooth representations of $p$-adic classical groups.
• To show that this algebra is isomorphic to a semidirect product of a Hecke algebra with parameters and a finite group algebra.
• To extend Bernstein's equivalence of categories to an explicit algebraic isomorphism for classical groups.
• To verify compatibility of the resulting algebraic structure with parabolic induction and Jacquet functors.
• To provide a concrete realization of the endomorphism algebra via bases over regular functions on orbit spaces.

Proposed method

• Utilizes Bernstein's equivalence between the category $Rep(^{W}\mathcal{O})$ and the category of right modules over $\mathrm{End}_G(i_P^G \mathrm{ind}_{M^1}^M E_1)$.
• Applies the theory of intertwining operators and their action on induced representations to analyze the endomorphism algebra.
• Constructs a basis for the endomorphism algebra over the ring of regular functions on the orbit space $\mathcal{O}$.
• Employs the action of the Weyl group and the finite group $R(\mathcal{O})$ to define the semidirect product structure.
• Uses the compatibility of intertwining operators with parabolic induction and Jacquet functors to verify functoriality.
• Relies on known parametrization of Hecke algebra parameters for classical groups, as established by Moeglin and Arthur.

Experimental results

Research questions

• RQ1 What is the explicit structure of the endomorphism algebra of the induced projective generator $i_P^G \mathrm{ind}_{M^1}^M E_1$ for $p$-adic classical groups?
• RQ2 Can the endomorphism algebra be realized as a semidirect product of a Hecke algebra with parameters and a finite group algebra?
• RQ3 How do the intertwining operators $T_w$ and $J_r$ interact under conjugation by elements of $R(\mathcal{O})$?
• RQ4 Is the resulting algebraic isomorphism compatible with parabolic induction and the Jacquet functor?
• RQ5 What are the precise parameters $q^{a_{s_\alpha}+b_{s_\alpha}}$ and $q^{a_i - b_i}$ in the Hecke algebra for classical groups?

Key findings

• The endomorphism algebra $\mathrm{End}_G(i_P^G E_{B_{\mathcal{O}}})$ is isomorphic to the semidirect product $\mathbb{C}[R(\mathcal{O})]\ltimes \mathrm{H}(\Sigma_{\mathcal{O}}, \{q^{a_{s_\alpha}+b_{s_\alpha}}\}, \{q^{a_i - b_i}\})$.
• The Hecke algebra parameters $q^{a_{s_\alpha}+b_{s_\alpha}}$ and $q^{a_i - b_i}$ are explicitly determined for classical groups, with known values from Arthur–Langlands theory.
• The action of $R(\mathcal{O})$ on the Weyl group elements satisfies $T_w J_r = J_r T_{r^{-1}wr}$, confirming compatibility with the semidirect product structure.
• The category $Rep(^{W}\mathcal{O})$ is equivalent to the category of right modules over the constructed semidirect product algebra.
• The isomorphism of categories is compatible with parabolic induction and the Jacquet functor, as verified via functoriality of intertwining operators.
• The results generalize to arbitrary connected reductive $p$-adic groups, though a twisted group algebra may be needed in the general case.