研究情報

Let $G$ be a reductive algebraic group over a field of characteristic $p > 0$ and consider the category of finite dimensional $G$-modules. Relative to some choice of positive roots for $G$ we fix a dominant weight $\lambda$. Then we shall introduce a quotient category consisting of modules whose composition factors are “close to $\lambda$” (we shall make the meaning of this precise in the talk). Our aim is now to study the effect of adding a $p$-multiple of a weight to $\lambda$. We show that under appropriate conditions this leaves the quotient category in question invariant. As a consequence we derive some invariance properties of composition factor multiplicities in Weyl modules with highest weights close to $\lambda$. We get a similar invariance property for the Weyl factor multiplicities in the corresponding indecomposable tilting modules. Our arguments also apply to quantum groups at roots of unity.

One of the key properties of the quantum group associated to a semisimple complex Lie algebra is the existence of its canonical basis defined independently by Lusztig and Kashiwara. This basis induces bases in all finite highest weight modules over the corresponding Lie algebra. The properties of the canonical basis are closely tied up in the properties of the associated combinatorics, arising from the various parametrization approaches. In this talk, I will give an explicit comparison between the parametrizations via Kashiwara operators and via generic extensions for quiver representations.

The existence of generic extensions for a Dynkin or cyclic quiver $Q$ gives rise to a graph $\textsf{G}$, called the generic extension graph. This graph has the same vertex set $\Lambda_{Q}$ as the crystal graph $\textsf{C}$ for the quantized enveloping algebra associated to $Q$. If $P(X)$ denotes the set of all paths between $0$ and $\lambda \in \Lambda_Q$ in graph $X$, we may define two maps $$ \wp_Q: P(\textsf{G}) \to \Lambda_{Q} \qquad\text{and}\qquad \kappa_{Q}: P(\textsf{C}) \to \Lambda_{Q}, $$ by sending a path to the endpoint $\lambda$. The two path sets can be naturally identified as the set $\Omega$ of all words on the alphabet $I$, the vertex set of $Q$. Thus, we obtain maps $\wp_{Q}, \kappa_{Q}: \Omega \to \Lambda_{Q}$. We prove that $\kappa_{Q}(w) \le \wp_{Q}(w)$ for all $w \in \Omega$ and $Q$ under the Bruhat type ordering on $\Lambda_{Q}$. Moreover, if $Q$ is a (finite or infinite) linear quiver, then the intersection of the fibres $\wp_{Q}^{-1}(\lambda)$ and $\kappa_{Q}^{-1}(\lambda)$ is non-empty for every $\lambda \in \Lambda_{Q}$. We will also show that this non-emptyness property fails for a cyclic quiver.

This is a joint work with Bangming Deng and Guanglian Zhang.

Let $H$ be an Iwahori–Hecke algebra associated with a finite Weyl group. In type $A_n$, it is known that the Kazhdan–Lusztig basis of $H$ is cellular in the sense of Graham–Lehrer. We report on a new, elementary proof of this result. Our methods also provide, for the first time, purely algebraic proofs of fundamental properties of Lusztig’s $a$-function (in type $A_{n}$). Similar arguments apply to type $B_{n}$, for a certain class of unequal parameters.

We develop the crystal basis theory for quantum generalized Kac–Moody algebras. We define the notion of crystal bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and prove the standard properties of crystal bases including the tensor product rule. We then prove that there exist crystal bases (and global bases) for $V(\lambda)$ $(\lambda \in P^{+})$ and $U_q^{-}(\mathfrak{g})$. We also introduce the notion of abstract crystals for quantum generalized Kac–Moody algebras and study their fundamental properties. Finally, we prove the crystal embedding theorem and give a characterization of the crystals $B(\infty)$ and $B(\lambda)$.

Let $\mathfrak{g}$ be a complex semisimple Lie algerba and $U_q$ the Drinfel’d–Jimbo quantisation over the field $\mathbb{C}(q)$ of rational functions in the indeterminate $q$. If $V$ and $V_q$ are corresponding irreducible modules for $\mathfrak{g}$ and $U_q$, it is known that in several cases, $\operatorname{End}_{U_q} V_q^{\otimes r}$ is a deformation of $\operatorname{End}_{\mathfrak{g}} V^{\otimes r}$, and both algebras have a cellular structure, which in principle permits one to study non-semisimple deformation of either.

We present a framework (“strongly multiplicity free” modules) where the endomorphism algebras are “generic” in the sense that in the classical (unquantised) case, they are quotients of Kohno’s infintesimal braid algebra $T_r$, while in the quantum case, they are quotients of the group ring $\mathbb{C}(q) B_r$ of the $r$-string braid group $B_r$. In addition to the well known cases above, these include the irreducible 7 dimensional module in type $G_2$ and arbitrary irreducibles for $\mathfrak{s} \mathfrak{l}_2$.

This is joint work with Ruibin Zhang.

Let $i$ be a homomorphism of the multiplicative group into a connected reductive algebraic group over $C$. Let $G^i$ be the centralizer of the image $i$. Let $LG$ be the Lie algebra of $G$ and let $L_nG$ ($n$ integer) be the summands in the direct sum decomposition of $LG$ determined by $i$. Assume that $n$ is not zero. For any $G^i$-orbit $O$ in $L_nG$ and any irreducible $G^i$-equivariant local system $L$ on $O$ we consider the restriction of some cohomology sheaf of the intersection cohomology complex of the closure of $O$ with coefficients in $L$ to another orbit $O’$ contained in the closure of $O$. For any irreducible $G^i$-equivariant local system $L’$ on $O’$ we would like to compute the multiplicity of $L’$ in that restriction. We present an algorithm which helps in computing that multiplicity.

This is a joint work with Bezrukavnikov and Rumynin. It is a step in Bezrukavnikov’s program of verification of Lusztig’s elaborate predictions of the numerical structure of representation theory for Lie algebras in positive characteristic. For a semisimple algebraic group in positive characteristic a version of the Beilinson–Bernstein formalism provides localization of Lie algebra representations to $D$-modules on the flag variety. In this setting, the analogue of the “Riemann–Hilbert” correspondence leads not to perverse sheaves but to twisted coherent sheaves on the cotangent bundle to the flag variety. This provides an exotic t-structure on coherent sheaves that corresponds to the standard one on representations. The fact that this $t$-structure is (roughly), defined over integers, is a categorical formulation of the independence of the representation theory on the characteristic.

(Affine) $W$-algebras are an interesting family of vertex algebras, which can be viewed as chiralization of the Kostant–Lynch theory. In this talk, we will describe their representation theory using the method of quantum reduction.

We give a purely combinatorial and non-recursive criterion for when Specht module $S^{(\lambda,\mu)}$ of Hecke algebra of type B has non-zero quotient $D^{(\lambda,\mu)}$. The proof uses Littelmann’s path model, in particular. This is a joint work with Kreiman and Tsuchioka.

We consider the equivariant $K$-thoery of affine flag manifolds with respect to the Borel action. The Scubert varities, the closure of Borel orbits, are finite-codimensional subvarieties of the flag manifolds, and they are normal and Cohen–Macaulay. We represent the class of the structure sheaf of the Scubert varieties by Laurent polynomials.

The Deligne–Langlands–Lusztig conjecture (proved by Kazhdan–Lusztig, Ginzburg) asserts that each simple module of an affine Hecke algebra corresponds to some geometric datum. An affine Hecke algebra of type $C$ admits a natural two-parameter deformation $\mathbb{H}$ (and this is best possible in some sense).

In this talk, we realize $\mathbb{H}$ as the equivariant $K$-group of a certain variety, which we refer as the exotic Steinberg variety. This enables us to present a Deligne–Langlands type classification of simple $\mathbb{H}$-modules when the values and ratios of deformation parameters are not too bad.

In the study of singularity theory, Kyoji Saito introduced a generalization of the theory of root systems (called elliptic root systems) and their Weyl groups (called elliptic Weyl groups). An elliptic Weyl group is the group generated by reflections attached to all roots in an elliptic root system. It is not Coxeter group, but its generators and relations are described by elliptic Dynkin diagrams due to Kyoji Saito.

In this talk we will talk on a $q$-analogue of elliptic Weyl groups called elliptic Hecke algebras. Firstly we will give a definition of elliptic Hecke algebra attaching to elliptic Dynkin diagrams. After that, we will discus on the relationship between double affine Hecke algebras and elliptic Hecke algebras.

Let $U_q(\mathfrak{g})$ be a quantum affine algebra with weight lattice $P$, and let $\lambda = \sum_{i \in I_{0}} m_i \varpi_{i} \in P$, with $m_{i} \in \mathbb{Z}_{\geq 0}$, be a level-zero dominant integral weight, where $\varpi_{i} \in P$, $i \in I_{0}$, are the level-zero fundamental weights. In this talk, we first give an explicit description of the crystal structure of the $P$-weighted crystal $\mathbb{B}(\lambda)$ of all Lakshmibai–Seshadri paths (LS paths) of shape $\lambda$, and then explain its relation to the crystal base of the extremal weight module of extremal weight $\lambda \in P$.

Second, we give a tensor product decomposition theorem for the $P_{cl}$-weighted crystal $\mathbb{B}(\lambda)_{cl}$, that $\mathbb{B}(\lambda)_{cl} \simeq \bigotimes_{i \in I_0} \mathbb{B}(\varpi)_{cl}^{\otimes m_{i}}$, and then explain its relation to the crystal base of the corresponding tensor product of level-zero fundamental representations $W(\varpi_{i})$, $i \in I_{0}$, of a quantum affine algebra $U_{q}^{\prime}(\mathfrak{g})$ with weight lattice $P_{cl}$.

Finally, we give an interpretation via LS paths of one-dimensional configuration sums associated to the level-zero fundamental representations $W(\varpi_{i})$, $i \in I_{0}$. In particular, in the case of type $A_{\ell}^{(1)}$, this gives a description of the Kostka-Foulkes polynomials in terms of LS paths.

This is a joint work with Satoshi Naito.

We study about highest weight representations of the double affine Hecke algebra (DAHA, or Cherednik algebra) of type $A$ and their relation with representations of the affine Lie algebra based on the conformal field theory (CFT) called Wess–Zumino–Witten model.

We will see that the CFT on the complex plane $\mathbb{C}$ gives a natural and explicit construction of a representation of the rational DAHA as a certain space of coinvariants constructed from highest weight representations of the affine Lie algebra.

Similarly, the CFT on the cylinder $\mathbb{C} \setminus \{0\}$ gives a construction of representations of the trigonometric DAHA (the degenerate DAHA). Our space of coinvariants coincides, after localization, with the vector bundle given by the CFT, and the construction above gives a functor between categories of highest weight representations of the affine Lie algebra and DAHA which is compatible with the ‘Knizhnik–Zamolodchikov functor’ between categories of representations of DAHA and the (affine) Iwahori–Hecke algebra studied by Ginzburg–Guay–Opdam–Rouquier and Varagnolo–Vasserot.

When a parameter (called the level) of the theory is generic, the functor from the plane model gives the classical Schur–Weyl reciprocity between the finite-dimensional Lie algebra and the symmetric group, and the functor from the cylinder model is reduced to the functor from the Bernstein–Gelfand–Gelfand category of the finite-dimensional Lie algebra to the category of finite-dimensional representations of the degenerate affine Hecke algebra. This functor has been introduced and studied by Arakawa–Suzuki–Tsuchiya.

We focus on integrable representations of the affine Lie algebra when the level is a positive integer. It turns out that irreducible integrable representations correspond through the functor to a special class of irreducible representations of DAHA which have a combinatorial description by ‘periodic’ standard tableaux and have a nice character formula described by level-restricted Kostka polynomials.

I will talk about my joint work with N. Xi (math.RT/0411304). According to Kazhdan–Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg’s triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. It is conjectured that the filtration is compatible with the Kazhdan–Lusztig basis of the Hecke algebra. In this talk we will give its proof in the case of type $A$.

We shall introduce several tropical $R$’s for affine geometric crystals explicitly. We can perform it by using certain involution on $M$-matrix of simply-laced affine Lie algebras.

We consider irreducible finite dimensional representations of quantum affine algebras. In 2000 I computed their $q$-characters, but the computation involved a combinatorial algorithm similar to the definition of Kazhdan–Lusztig polynomials. Thus the answers were not explicit. In this talk I will give some examples of explicit formulas and also some examples of crystal bases.

In my talk I would like to talk about representation theory of symmetric groups, Iwahori–Hecke algebras of type A and their $q$-Schur algebras at roots of unity.

The focus here is to compare module categories of Hecke algebras in different roots of unity. The result even works for any blocks at any roots of unity in postive characteristics with some suitable assumptions and has some applications for James’ conjecture which is consistent with Lusztig’s character formula conjecture for type A.

Our method for the new results is based on the knowledge of on so-called Rouquier block algebras and derived equivalences of blocks algebras in Hecke algebras in terms of Chuang–Rouquier’s $\mathfrak{sl}_2$-categorifications and perverse Morita equivalences.