arXiv subjects
Publications and source records attributed to Xuhua He.
In this paper, we study the relation between the cocenter $\overline{\tilde \ch_0}$ and the finite dimensional representations of an affine $0$-Hecke algebra $\tilde \ch_0$. As a consequence, we obtain a new criterion on the supersingular modules: a (virtual) module of $\tilde \ch_0$ is supersingular if and only if its character vanishes on the non-supersingular part of $\overline{\tilde \ch_0}$.
In this paper, we study the relation between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual object in representation theory, the rigid quotient of the Grothendieck group of finite dimensional representations.
Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter function) has been studied by many mathematicians, including Kazhdan-Lusztig \cite{KL}, Ginzburg \cite{CG}, Lusztig \cite{L1}, Reeder \cite{Re}, Opdam-Solleveld \cite{OS}, Kato \cite{Kat}, etc. The approaches are either geometric or analytic. In this note, we'll discuss a different route, via the so-called "cocenter-representation duality", to study finite dimensional representations of affine Hecke algebras (for arbitrary isogeny class and for a generic complex parameter). This route is more algebraic, and allows us to work with complex parameters, instead of equal parameters or positive parameters. We also expect that it can be eventually applied to the "modular case" (for representations over fields of positive characteristic and for parameter equal to a root of unity).
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.
This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in the study of the cocenters and class polynomials of affine Hecke algebras $\ch$ and their relation to affine Deligne-Lusztig varieties. Let $w$ be a $P$-alcove element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study the image of $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theorem for affine Deligne-Lusztig varieties. As a consequence, we present a new proof of \cite{GHKR} and \cite{GHN} on the emptiness pattern of affine Deligne-Lusztig varieties.
This paper studies affine Deligne-Lusztig varieties $X_{\tw}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tw}(b)$ for a minimal length element $\tw$ in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in \cite{HL} to the affine case. We then provide a reduction method that relates the structure of $X_{\tw}(b)$ for arbitrary elements $\tw$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of Görtz, Haines, Kottwitz and Reuman in \cite{GHKR}.
We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen Macaulay.
This note is based on my talk at ICCM 2013, Taipei. We give an exposition of the group-theoretic method and recent results on the questions of non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties.
Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $.
We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover $\wti W$, a certain double cover of the Weyl group $W$, and an extended Dirac operator for graded Hecke algebras. Our approach leads to a new and uniform construction of the irreducible genuine $\wti W$-characters. In the process, we give a construction of the action by an outer automorphism of the Dynkin diagram on the cohomology groups of Springer theory, and we also introduce a $q$-elliptic pairing for $W$ with respect to the reflection representation $V$. These constructions are of independent interest. The $q$-elliptic pairing is a generalization of the elliptic pairing of $W$ introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product $A_W=\bC[W]\ltimes S(V)$.
We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in Görtz et al. (2010). The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.
We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (Density theorem) and that the image is the space of good forms (trace Paley-Wiener theorem).
Let $W_a$ be an affine Weyl group and $η:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $η(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold for twisted conjugacy classes.
Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that $T_{w_\co}$, where $\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}.
In this notes, we will give an exposition of some results on the method of partial conjugation action. We first discuss the partial conjugation action of a parabolic subgroup of a Coxeter group. We then discuss some applications to Lusztig's $G$-stable pieces and its affine generalization. We also discuss some recent work on the $\s$-conjugacy classes of loop groups and affine Deligne-Lusztig varieties.
We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group $W$ have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in $W$.
Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and Y' is finite.
Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the method of Frobenius splitting and of global F-regularity.