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Xuhua He

Publications and source records attributed to Xuhua He.

At least 91 records · Page 5Linked to original sources
arXiv2010

Dimensions of affine Deligne-Lusztig varieties in affine flag varieties

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic $σ$-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic $σ$-conjugacy class is the class of $b=1$, we also prove the dimension formula predicted in op. cit. in almost all cases.

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arXiv2010

Minimal length elements in conjugacy classes of extended affine Weyl group

We study the minimal length elements in an integral conjugacy class of a classical extended affine Weyl group and we show that these elements are quite "special" in the sense of Geck and Pfeiffer \cite{GP93}. We also discuss some application on extended affine Hecke algebras and loop groups.

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arXiv2010

Closure of Steinberg fibers and affine Deligne-Lusztig varieties

We discuss some connections between the closure $\bar F$ of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties $X_w(1)$ in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of $X_w(1)$ if the translation part of $w$ is quasi-regular. As a by-product, we give a new proof of the explicit description of $\bar F$, first obtained in \cite{H1}.

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arXiv2009

On some partitions of an affine flag variety

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w(b)$ introduced by Lusztig in \cite{L1} as part of the definition of character sheaves. Among other things, we give a formula for the dimension of affine Deligne-Lusztig varieties for classical loop groups in terms of degrees of class polynomials of extended affine Hecke algebra. We also prove that any simple $GL_n(\FF_q((\e)))$-module occurs as a subquotient of the cohomology of affine Deligne-Lusztig variety $X_w(1)$ for some $w$ in the extended affine Weyl group $\ZZ^n \rtimes S_n$ must occurs for some $w$ in the finite Weyl group $S_n$. Similar result holds for $Sp_{2n}$.

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arXiv2009

On intersections of certain partitions of a group compactification

Let $G$ be a connected semi-simple algebraic group of adjoint type over an algebraically closed field, and let $\overline{G}$ be the wonderful compactification of $G$. For a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we look at intersections of Lusztig's $G$-stable pieces and the $B^-\times B$-orbits in $\overline{G}$, as well as intersections of $B \times B$-orbits and $B^- \times B^-$-orbits in $\overline{G}$. We give explicit conditions for such intersections to be non-empty, and in each case, we show that every non-empty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the non-empty intersections form a strongly admissible partition of $\overline{G}$.

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arXiv2009

Semi-stable locus of a group compactification

In this paper, we consider the diagonal action of a connected semisimple group of adjoint type on its wonderful compactification. We show that the semi-stable locus is a union of the $G$-stable pieces and we calculate the geometric quotient.

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arXiv2009

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the ``ordinary'' restriction of a character sheaf on the compactification to a boundary piece inside the semi-stable locus is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer \cite{Sp2}. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture \cite[12.6]{L3} inside the semi-stable locus of the wonderful compactification.

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arXiv2009

A subalgebra of 0-Hecke algebra

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.

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arXiv2008

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.

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arXiv2008

Frobenius splitting and geometry of $G$-Schubert varieties

Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\mathcal R$-Schubert varieties.

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arXiv2007

On the affineness of Deligne-Lusztig varieties

We prove that the Deligne-Lusztig variety associated to minimal length elements in any $\d$-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in \cite{OR}.

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arXiv2006

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.

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arXiv2006

Character sheaves on certain spherical varieties

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.

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arXiv2006

The $G$-stable pieces of the wonderful compactification

Let $G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $\bar{G}$ of $G$ into finite many $G$-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We will show that the closure is a disjoint union of some $G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $G$-orbits.

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arXiv2006

The character sheaves on the group compactification

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in \cite{L3}. We also prove some properties of the character sheaves on the group compactification.

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