[Paper Review] Hecke algebras for inner forms of p-adic special linear groups

Anne‐Marie Aubert, Paul Baum+2 more|arXiv (Cornell University)|Jun 26, 2014

Advanced Algebra and Geometry16 references4 citations

TL;DR

This paper constructs Hecke algebras for Bernstein components of inner forms of $p$-adic special linear groups $\mathrm{SL}_n(F)$ via restriction from inner forms of $\mathrm{GL}_n(F)$. By leveraging types for $\mathrm{GL}_n(F)$ and descending to the derived group $G^\sharp$, it exhibits an explicit algebra whose module category is equivalent to the category of complex smooth $G^\sharp$-representations for each L-indistinguishable packet of Bernstein components. The key result is that these Hecke algebras are similar to affine Hecke algebras of type $A$ but are not always Morita equivalent to a crossed product of such an algebra with a finite group.

ABSTRACT

Let F be a non-archimedean local field and let $G^\sharp$ be the group of F-rational points of an inner form of $SL_n$. We study Hecke algebras for all Bernstein components of $G^\sharp$, via restriction from an inner form G of $GL_n (F)$. For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth $G^\sharp$-representations. This algebra comes from an idempotent in the full Hecke algebra of $G^\sharp$, and the idempotent is derived from a type for G. We show that the Hecke algebras for Bernstein components of $G^\sharp$ are similar to affine Hecke algebras of type A, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

Motivation & Objective

• To understand the representation theory of inner forms of $p$-adic $\mathrm{SL}_n(F)$ using Hecke algebras associated to Bernstein components.
• To construct a Morita equivalence between the category of smooth complex $G^\sharp$-representations and a module category over an explicit Hecke algebra.
• To show that the resulting Hecke algebras for $G^\sharp$ are structurally similar to affine Hecke algebras of type $A$, yet not always Morita equivalent to a crossed product of such an algebra with a finite group.
• To establish a descent procedure from $\mathrm{GL}_n(F)$-types to $\mathrm{SL}_n(F)$-types via idempotents in the Hecke algebra of the derived group.

Proposed method

• Use of types for $\mathrm{GL}_n(F)$-representations, specifically the construction of types for Bernstein components by Sécherre–Stevens.
• Construction of a particular idempotent in the full Hecke algebra of $G^\sharp$ derived from a type in $G = \mathrm{GL}_m(D)$, where $D$ is a division algebra over $F$.
• Descent of the Hecke algebra via restriction from $G$ to its derived group $G^\sharp$, using the reduced norm map and the structure of the intermediate group $H^\sharp = G^\sharp Z(G)$.
• Application of Bernstein's theory of Bernstein blocks and the use of the Bernstein torus $T_{\mathfrak{s}}$ and finite Weyl group $W_{\mathfrak{s}}$ to classify components.
• Use of projective normalizers and intertwining operators to relate representations across $G$ and $G^\sharp$, particularly via $I(\gamma, \pi)$ and $X^G({\mathfrak{s}}/\lambda)$.
• Explicit computation of the Hecke algebra $\mathcal{H}(G^\sharp)^{\mathfrak{s}}$ as a product of algebras $\mathcal{H}(G^\sharp)^{\mathfrak{t}^\sharp}$ over $\mathfrak{t}^\sharp \prec \mathfrak{s}$, with the final algebra arising from an idempotent in $\mathcal{H}(G^\sharp)$.

Experimental results

Research questions

• RQ1 How can the category of smooth complex representations of an inner form $G^\sharp$ of $\mathrm{SL}_n(F)$ be described using Hecke algebras?
• RQ2 What is the structure of the Hecke algebra $\mathcal{H}(G^\sharp)^{\mathfrak{s}}$ associated to a Bernstein component $\mathfrak{s}$ of $G^\sharp$?
• RQ3 To what extent are these Hecke algebras for $G^\sharp$ Morita equivalent to affine Hecke algebras of type $A$?
• RQ4 Can the representation theory of $G^\sharp$ be reduced to that of $G = \mathrm{GL}_m(D)$ via restriction and idempotent descent?
• RQ5 How do L-indistinguishable Bernstein components of $G^\sharp$ relate to one another in terms of their Hecke algebras?

Key findings

• The Hecke algebra $\mathcal{H}(G^\sharp)^{\mathfrak{s}}$ for a Bernstein component $\mathfrak{s}$ of $G^\sharp$ is Morita equivalent to the module category of a specific algebra constructed from an idempotent in $\mathcal{H}(G^\sharp)$.
• This algebra arises from a type for $G = \mathrm{GL}_m(D)$, and the idempotent is constructed via descent from the Hecke algebra of $G$ to $G^\sharp$.
• The resulting Hecke algebra for $G^\sharp$ is structurally similar to an affine Hecke algebra of type $A$, but in general is not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.
• The category $\mathrm{Rep}^\mathfrak{s}(G^\sharp)$ decomposes as a product of Bernstein blocks $\mathrm{Rep}^{\mathfrak{t}^\sharp}(G^\sharp)$ indexed by $\mathfrak{t}^\sharp \prec \mathfrak{s}$, and the corresponding Hecke algebras $\mathcal{H}(G^\sharp)^{\mathfrak{t}^\sharp}$ are maximal indecomposable subalgebras of $\mathcal{H}(G^\sharp)^{\mathfrak{s}}$.
• For any L-packet of Bernstein components of $G^\sharp$, the associated Hecke algebra is explicitly described and yields a Morita equivalence to the category of smooth $G^\sharp$-representations in that packet.
• The construction is validated through examples, including cases where types are conjugate in $G^\sharp$ but not in $G^1$, demonstrating the necessity of the full group structure in the descent process.