arXiv subjects

Stefan Dawydiak

Publications and source records attributed to Stefan Dawydiak.

8 recordsLinked to original sources

arXiv2026

On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrm{GL}_n$

We study Braverman-Kazhdan's asymptotic Hecke algebra $\mathcal{J}(G)$ for inner forms $G$ of $p$-adic $\mathrm{GL}_n$. We show that $\mathcal{J}(G)$ and the property for a $G$-representation to extend to a $\mathcal{J}(G)$-module are defined over $\overline{\mathbb{Q}_\ell}$, and hence make sense in the context of the categorical local Langlands correspondence. We show a rudimentary form of compatibility with Hecke operators, allowing us discuss stalks of sheaves on $\mathrm{Bun}_n$ corresponding to the trivial vector bundles on the stack of $L$-parameters, in particular the Whittaker sheaf, in terms of $\mathcal{J}(\mathrm{GL}_n)$-modules. We provide explicit formulas in terms of reductive centralizer of $L$-parameters for many functions in $\mathcal{J}(G)$, and show that $\mathcal{J}(G)$ has the same Hochschild homology as $C_c^\infty(G)$, and that the Kazhdan-Lusztig bijection appears in the isomorphism. We proceed via Bushnell-Kutzko and Sécherre-Stevens types, generalizing a theorem of Suzuki for $\mathrm{GL}_n$ for which we provide a proof.

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arXiv2026

A positivity property in the based ring of the lowest two-sided cell

Let $W_{\mathrm{aff}}$ be an extended affine Weyl group and $\mathbf{H}$ and $J$ be the corresponding affine and asymptotic Hecke algebras with standard bases $\{T_x\}$ and $\{t_w\}$, respectively. Viewing $J$ as a subalgebra of the $\mathbf{q}^{-\frac{1}{2}}$-adic completion of $\mathbf{H}$, we give formulas for the coefficient of $T_x$ in $t_w$ for various $x$ and $w$ in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing $w$. In particular our results hold for the canonical left cell. For such $w$ we also define a seemingly new positive basis for the corresponding subring of $J$. For $\mathrm{GL}_n$, we give partial results for some other cells.

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arXiv2026

Standard modules of affine Hecke algebras

Let $G$ be a connected reductive group defined and split over a non-archimedean local field $F$. We give a new geometric proof of a special case of a recent theorem of Solleveld. Namely, we show that the class of standard Iwahori-spherical $G(F)$-representations, a notion a priori dependent on the coefficient field being the complex numbers, is actually defined over $\overline{\mathbb{Q}_\ell}$. An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially square-integrable representations is also defined over $\overline{\mathbb{Q}_\ell}$. As an application of our main result, we give a local proof of this theorem for inner forms of $\mathrm{GL}_n$, as well as showing that standard representations of these groups are defined over $\overline{\mathbb{Q}_\ell}$.

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arXiv2025

The asymptotic Hecke algebra and rigidity

We reprove the surjectivity statement of Braverman-Kazhdan's spectral description of Lusztig's asymptotic Hecke algebra $J$ in the context of $p$-adic groups. The proof is based on Bezrukavnikov-Ostrik's description of $J$ in terms of equivariant $K$-theory. As a porism, we prove that the action of $J$ extends from the non-strictly positive unramified characters to the complement of a finite union of divisors, and that the trace pairing between the Ciubotaru-He rigid cocentre of an affine Hecke algebra with equal parameters and the rigid quotient of its Grothendieck group is perfect whenever the parameter $q$ is not a root of the Poincaré polynomial of the finite Weyl group. Without recourse to $K$-theory, we prove a weak version of Xi's description of $J$ in type $A$. As an application of relationship between $J$ and the rigid cocentre, we prove that the formal degree of a unipotent discrete series representation of a connected reductive $p$-adic group $G$ with a split inner form has denominator dividing the Poincaré polynomial of the Weyl group of $G$. Additionally, we give formulas for $t_w$ in terms of inverse and spherical Kazhdan-Lusztig polynomials for $w$ in the lowest cell.

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arXiv2025

Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula

Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be the corresponding affine Hecke algebra over the ring $\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing $J$ as a subalgebra of the $(\mathbf{q}^{-\frac{1}{2}})$-adic completion of $\mathbf{H}$ via Lusztig's map $ϕ$, we use Harish-Chandra's Plancherel formula for $p$-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf{q}$, with denominator depending only on the two-sided cell containing $w$, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known. Along the way, we show that upon specializing $\mathbf{q}=q>1$, the map from $J$ to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter.

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arXiv2024

A coherent categorification of the based ring of the lowest two-sided cell

We give a partial coherent categorification of $J_0$, the based ring of the lowest two sided cell of an affine Weyl group, equipped with a monoidal functor from the category of coherent sheaves on the derived Steinberg variety. We show that our categorification acts on natural coherent categorifications of the Iwahori invariants of the Schwartz space of the basic affine space. In low rank cases, we construct complexes that lift the basis elements $t_w$ of $J_0$ and their structure constants.

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arXiv2023

On the structure of the affine asymptotic Hecke algebras

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also show that Lusztig's homomorphism from the affine Hecke algebra to the asymptotic affine Hecke algebra induces an isomorphism on cocenters and discuss the relation of the above central extensions to the structure of the cocenter.

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arXiv2021

On Lusztig's asymptotic Hecke algebra for $\mathrm{SL}_2$

Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis $\{t_w\}$ of $J$, and further invert the canonical isomorphism between the completions of $H$ and $J$, obtaining explicit formulas for the each basis element $t_w$ in terms of the basis $\{T_w\}$ of $H$. We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that $J$ acts on the Schwartz space of the basic affine space of $\mathrm{SL}_2$, and produce some formulas for this action.

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