arXiv subjects
Publications and source records attributed to Toshiaki Shoji.
We construct a monomial basis of a quantum affine algebra of simply-laced type, associated to the PBW basis of Beck-Nakajima. We show that there exists a simple algorithm of computing canonical basis in terms of the monomial basis. We dsicuss the relations of the canonical basis obtained from this PBW basis with Lusztig's canonical basis constructed by using the geometry of quivers.
Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. Let $P$ be the transition matrix between the canonical basis and a PBW basis of ${\mathbf U}_q^-$. In the case ${\mathbf U}_q^-$ is symmetric, Antor gave a simple algorithm of computing $P$ by making use of monomial bases. By the folding theory, ${\mathbf U}_q^-$ (symmetric, with a certain automorphism) is related to a quantum group $\underline{\mathbf U}_q^-$ of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for ${\mathbf U}_q^-$ and for $\underline{\mathbf U}_q^-$.
Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis. Our construction is elementary, in the sense that we don't appeal to Lusztig's geometric theory of canonical bases, nor to Kashiwara's theory of crystal bases.
In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent class. The existence of split elements is reduced to the case where $G$ is a simply connected, almost simple group. We show, in the case of classical groups, split elements exist, which is a refinement of previous results. In the case of exceptional groups, we show the existence of split elements, possibly except one class for $G$ of type $E_7$.
Let ${\mathbf U}^-_q$ be the negative half of the quantum group associated to a Kac-Moody algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the quantum group obtained by a folding of ${\mathfrak g}$. Let ${\mathbf A} = {\mathbf Z}[q,q^{-1}]$. McNamara showed that $\underline{\mathbf U}^-_q$ is categorified over a certain extenion ring $\widetilde{\mathbf A}$ of ${\mathbf A}$, by uing the folding theory of KLR algebras. He posed a question whether $\widetilde{\mathbf A}$ coincides with ${\mathbf A}$ or not. In this paper, we give an affirmative answer for this problem.
Let ${\mathbf U}^-_q$ be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra ${\mathfrak g}^σ$ obtained from an admissible diagram automorphism $σ$ on ${\mathfrak g}$. Lusztig consructed the canonical basis ${\mathbf B}$ of ${\mathbf U}_q^-$ and the canonical signed basis $\underline{\widetilde{\mathbf B}}$ of $\underline{\mathbf U}_q^-$ by making use of the geometric theory of quivers. He proved that there is a natural bijection $\widetilde{\mathbf B}^σ \to \widetilde{\underline{\mathbf B}}$. In this paper, assuming the existence of the canonical basis ${\mathbf B}$ of ${\mathbf U}_q^-$, we construct the canonical signed basis $\widetilde{\underline{\mathbf B}}$ of $\underline{\mathbf U}_q^-$, and a natural bijection $\widetilde{\mathbf B}^σ \to \widetilde{\underline{\mathbf B}}$ by an elementary method.
Let ${\mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $σ$ the algebra automorphism on ${\mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{\mathbf U}_q^-$ be the quantum algebra obtained from $σ$, and $\widetilde{\mathbf B}$ (resp. $\widetilde{\underline{\mathbf B}}$) the canonical signed basis of ${\mathbf U}_q^-$ (resp. $\underline{\mathbf U}_q^-$). Assume that ${\mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection $\widetilde{\mathbf B}^σ \simeq \widetilde{\underline{\mathbf B}}$ in the case where $σ$ is admissible. In this paper, we show that such a bijection exists even if $σ$ is not admissible, possibly except some small rank cases.
Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.
Let $G = GL_N$ over an algebraically closed field of odd characteristic, and $θ$ an involutive automorphism on $G$ such that $H = (G^θ)^0$ is isomorphic to $SO_N$. Then $G^{ιθ} = \{ g \in G \mid θ(g) = g^{-1} \}$ is regarded as a symmetric space $G/G^θ$. Let $G^{ιθ}_{uni}$ be the set of unipotent elements in $G^{ιθ}$. $H$ acts on $G^{ιθ}_{uni}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{ιθ}_{uni}$ and irreducible representations of various symmetric groups.
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $σ$ on ${\mathfrak g}$. Let ${\mathbf B}^σ$ be the set of $σ$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^σ \simeq \underline{\mathbf B}$ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where ${\mathfrak g}$ is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence between PBW-bases, by using a new type of PBW-bases of ${\mathbf U}_q^-$ obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.
In this paper, we consider the set of r-symbols in a full generality. We construct Hall-Littlewood functions and Kostka functions associated to those r-symbols. We also discuss a multi-parameter version of those functions. We show that there exists a general algorithm of computing multi-parameter Kostka functions. As an application, we show that the generalized Green functions of symplectic groups can be described combinatorially in terms of those (one-parameter) Kostka functions.
Let $U^-_q = U^-_q(\mathfrak g)$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak g$, and $σ: \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak g^σ$ be the fixed point subalgebra of $\mathfrak g$, and put $\underline U^-_q = U^-_q(\mathfrak g^σ)$. Let $B$ be the canonical basis of $U_q^-$ and $\underline B$ the canonical basis of $\underline U_q^-$. $σ$ induces a natural action on $B$, and we denote by $B^σ$ the set of $σ$-fixed elements in $B$. Lusztig proved that there exists a canonical bijection $B^σ \simeq \underline B$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.
This is a survey article on the Springer correspondence for symmetric spaces. We discuss various generalization of the theory of the Springer correspondence for reductive groups to symmetric spaces and exotic symmetric spaces associated to classical groups.
Let $G = GL(V)$ for an N-dimensional vector space $V$ over an algebraically closed field k, and $G^θ$ the fixed point subgroup of $G$ under an involution $θ$ on $G$. In the case where $G^θ = O(V)$, the generalized Springer correspondence for the unipotent variety of the symmetric space $G/G^θ$ was studied by last two authors, under the assumption that ch k is odd. The definition of $θ$, and of the associated symmetric space given there make sense even if ch k = 2. In this paper, we discuss the Springer correspondence for those symmetric spaces of even characteristic. We show that if N is even, the Springer correspondence is reduced to that of symplectic Lie algebras in ch k = 2, which was determined by Xue. While if N is odd, we show that a very similar phenomenon as in the case of exotic symmetric space of level 3 appears.
Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G = GL(V)$. For a positive integer $r$, we consider the variety $X_{uni} = G_{uni} \times V^{r-1}$, on which $G$ acts diagonally. $X_{uni}$ is the "unipotent part" of the enhanced variety of level $r$. $X_{uni}$ is partitioned into finitely many pieces $X_λ$ labelled by $r$-partitions $λ$ of $n$, and we consider the intersection cohomology $IC_λ$ associated to $X_λ$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $IC_λ$ satisfy certain orthogonality relations, which are very close to the equations characterizing the Kostka functions indexed by (a pair of) $r$-partitions. Using this we show, in some special cases, that the Kostka functions can be described in terms of those intersection cohomology, which is a (partial) generalization of the known results for the case $r = 1, 2$.
Kostka functions $K^{\pm}_{λ, μ}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by $r$-partitions $λ, μ$ and a sign $+, -$. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov defined alternate functions $K_{λ,μ}(t)$ by using an analogue of Lusztig's partition function, and showed that $K_{λ,μ}(t)$ are polynomials in $t$ with non-negative integer coefficients. They conjecture that their $K_{λ,μ}(t)$ coincide with $K^-_{λ,μ}(t)$. In this paper, we show that their conjecture holds. We also discuss a multi-variable version of Kostka functions.
This paper is a survey on the topics concerning the Springer correspondence related to the varieties such as the enhanced variety or the exotic symmetric space. We explain in the case of exotic symmetric space of higher level, the complex reflection group $S_n \ltimes (\BZ/r\BZ)^n$ appears naturally in the framework of the Springer correspondence.
Kostka functions $K^{\pm}_{λ, μ}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $λ, μ$ of $r$-partitions and a sign $+, -$. It is expected that there exists a close connection between those Kostka functions and the intersection cohomology associated to the enhanced variety $X$ of level $r$. In this paper, we study combinatorial properties of Kostka functions by making use of the geometry of $X$. In particular, we show that if $μ$ is of the form $μ= (-,\dots, -, ξ)$ and $λ$ is arbitrary, $K^-_{λ, μ}(t)$ has a Lascoux-Schützenberger type combinatorial description.