Abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

Generalizing the dihedral picture for G(M,M,2) , we construct Hecke algebras (and present a strategy for constructing Hecke categories) and asymptotic counterparts. We think of these as associated with the complex reflection group G(M,M,N) .

In a recent joint paper with S. Sahi and V. Venkateswaran (2025), families of actions of the double affine Hecke algebra on spaces of quasi-polynomials were introduced. These so-called quasi-polynomial representations led to the introduction of quasi-polynomial extensions of the nonsymmetric Macdonald polynomials, which reduce to metaplectic Iwahori-Whittaker functions in the $\mathfrak{p}$-adic limit. In this paper, these quasi-polynomial representations are extended to Sahi's 5-parameter double affine Hecke algebra, and the quasi-polynomial extensions of the nonsymmetric Koornwinder polynomials are introduced.

Abstract Let $W_{\mathrm {aff}}$ 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 $\phi $ , 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.

abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

We study the space, R m R_m , of m m -symmetric functions consisting of polynomials that are symmetric in the variables x m + 1 , x m + 2 , x m + 3 , … x_{m+1},x_{m+2},x_{m+3},\dots but have no special symmetry in the variables x 1 , … , x m x_1,\dots ,x_m . We obtain m m -symmetric Macdonald polynomials by t t -symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of R m R_m . We define m m -symmetric Schur functions through a somewhat complicated process involving their dual basis, multi-Schur functions, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the m m -symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of m m -symmetric Schur functions. We obtain relations on the ( q , t ) (q,t) -Koska coefficients K Ω Λ ( q , t ) K_{\Omega \Lambda }(q,t) in the m m -symmetric world, and show in particular that the usual ( q , t ) (q,t) -Koska coefficients are special cases of the K Ω Λ ( q , t ) K_{\Omega \Lambda }(q,t) ’s. Finally, we show that when m m is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials.

We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac–Moody group of type A n-1 and vertex-labeled trees on n vertices. By an argument of Galashin, Lam, and Williams using Opdam’s trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on n vertices.

We compare two generalisations of the notion of hook lengths for partitions. We apply this in the context of the modular representation theory of Arike–Koike algebras. We show that the Schur element of a simple module is divisible by the Schur element of the associated (generalised) core. In the case of Hecke algebras of type A, we obtain an even stronger result: the Schur element of a simple module is equal to the product of the Schur element of its core and the Schur element of its quotient.

We study the geometry and topology of Δ-Springer varieties associated with two-row partitions. These varieties were introduced in recent work by Griffin–Levinson–Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund–Remmel–Wilson. We provide an explicit and combinatorial description of the irreducible components of the two-row Δ-Springer variety and compare it to the ordinary two-row Springer fiber as well as Kato’s exotic Springer fiber corresponding to a one-row bipartition. In addition to that, we extend the action of the symmetric group on the homology of the two-row Δ-Springer variety to an action of a degenerate affine Hecke algebra and relate this action to a 𝔤𝔩 2 -tensor space.

ABSTRACT We prove a number of results about the structure of the standard representation of the stable limit double affine Hecke algebra (DAHA). More precisely, we address the triangularity, spectrum, and eigenfunctions of the limit Cherednik operators, and construct several bases of Poincaré-Birkhoff-Witt (PBW) type for the stable limit DAHA. We establish a remarkable triangularity property concerning the contribution of certain special elements of the PBW basis of a finite rank DAHA of high enough rank to the PBW expansion of a PBW basis element of the stable limit DAHA. The triangularity property implies the faithfulness of the standard representation. This shows that the algebraic structure defined by the limit operators associated to elements of the finite rank DAHAs is precisely the stable limit DAHA.

The main question we address in this paper is: How much does the representation theory of the p-adic group GL n ( D ) depend on the p-adic division algebra D ? Here D is a central division algebra defined over a p-adic field F. Using Bushnell-Kutzko theory of types and Sécherre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category R ( GL n ( D ) ) of smooth complex representations of GL n ( D ) do not depend on D . In particular, when n = 1 or 2, the category R ( GL n ( D ) ) does not depend on D .

We introduce a class of polynomials that we call fused Specht polynomials and use them to characterize irreducible representations of the fused Hecke algebra with parameter q = − 1 q=-1 in the space of polynomials. We apply the fused Specht polynomials to construct a basis for a space of holomorphic (chiral) conformal blocks with central charge c = 1 c=1 which are degenerate at each point. In conformal field theory, this corresponds to all primary fields having conformal weight in the Kac table. The associated correlation functions are expected to give rise to conformally invariant boundary conditions for the Gaussian free field, which has also been verified in special cases.

Let W W be an affine Weyl group, and let k \Bbbk be a field of characteristic p > 0 p>0 . The diagrammatic Hecke category D \mathcal {D} for W W over k \Bbbk is a categorification of the Hecke algebra for W W with rich connections to modular representation theory. We explicitly construct a functor from D \mathcal {D} to a matrix category which categorifies a recursive representation ξ : Z W → M p r ( Z W ) \xi : \mathbb {Z}W \rightarrow M_{p^r}(\mathbb {Z}W) , where r r is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are “smaller” by a factor of p p . It also explains the presence of self-similarity in the p p -canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the p p -canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent p p -canonical tilting character formula due to Achar–Makisumi–Riche–Williamson.

Abstract We describe the center of the Hecke algebra of a type attached to a Bernstein block under a certain natural hypothesis. In particular, our hypothesis holds for the types considered by Kim–Yu and Fintzen and thus holds, under a mild tameness hypothesis, for all Bernstein blocks. We use our results to give a description of the Bernstein center of the Hecke algebra ℋ ⁢ ( 𝐆 ⁢ ( F ) , K ) {\mathcal{H}({\bf G}(F),K)} when K belongs to a nice family of compact open subgroups of 𝐆 ⁢ ( F ) {{\bf G}(F)} (which includes the Moy–Prasad filtrations of an Iwahori subgroup) via the theory of types.

We extend the recently-introduced weak Bruhat interval modules of the type A $0$-Hecke algebra to all finite Coxeter types. We determine, in a type-independent manner, structural properties for certain general families of these modules, with a primary focus on projective covers and injective hulls. We apply this approach to recover a number of results on type A $0$-Hecke modules in a uniform way, and obtain some additional results on recently-introduced families of type A $0$-Hecke modules.

Abstract We prove that all wild blocks of type Hecke algebras with quantum characteristic — that is, blocks of weight at least 2 — are strictly wild, with the possible exception of the weight 2 Rouquier block for . As a corollary, we show that for , all wild blocks of the ‐Schur algebras are strictly wild, without exception.

Representation theory of p -adic groups is a topic at a crossroads. It links among others to harmonic analysis, algebraic geometry, number theory, Lie theory, and homological algebra. The atomic objects in the theory are supercuspidal representations. Most of their aspects have a strong arithmetic flavour, related to Galois groups of local fields. All other representations are built from these atoms by parabolic induction, whose study involves Hecke algebras and complex algebraic geometry. In the local Langlands program, connections between various aspects of representations of p -adic groups have been conjectured and avidly studied. This workshop brought together mathematicians from various backgrounds, who hold the promise to contribute to the solution of open problems in the representation theory of p -adic groups. Topics included explicit local Langlands correspondences, Hecke algebras for Bernstein components, harmonic analysis, covering groups and \ell -modular representations of reductive p -adic groups.

Recipes to compute the algebraic K-theory of Hecke algebras of reductive p-adic groups

We introduce a general method for constructing modules for 0-Hecke algebras and supermodules for 0-Hecke–Clifford algebras from diagrams of boxes in the plane, and give formulas for the images of these modules in the algebras of quasisymmetric functions and peak functions under the relevant characteristic map. As initial applications, we resolve a question of Jing and Li (J Combin Theory Ser A 135:268–290, 2015), introduce a new basis of the peak algebra analogous to the quasisymmetric Schur functions, uncover a new connection between Schur Q-functions and quasisymmetric Schur functions, give a representation-theoretic interpretation of families of tableaux used in constructing certain functions in the peak algebra, and establish a common framework for known 0-Hecke module interpretations of bases of quasisymmetric functions.

Representations of Hecke algebras and Markov dualities for interacting particle systems