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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 $\sigma$ on ${\mathfrak g}$. Let ${\mathbf B}^{\sigma}$ be the set of $\sigma$-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 ${\mathbf B}^{\sigma} \simeq \underline{\mathbf B}$ 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., Comment: 43 pages

Using a dataset that recorded a large number of investment transactions in China from 1991 to 2018, we examine the statistical properties of the Chinese venture capital (VC) syndication network. Our main findings are as follows. First, the number of investment transactions sharply increased after 2014. Second, more than half of the VC firms are located in Beijing, Shanghai, and Shenzhen. Third, the degree distribution becomes roughly straight on a log–log plot. Fourth, the hypothesis that the degree distribution follows a power-law distribution is not rejected for 2015 and 2016. Fifth, better connected VC firms increase their connectivity faster, which suggests the existence of preferential attachment.

Let ${\mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $\sigma$ the algebra automorphism on ${\mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{\mathbf U}_q^-$ be the quantum algebra obtained from $\sigma$, 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 $\widetilde{\mathbf B}^{\sigma} \simeq \widetilde{\underline{\mathbf B}}$ in the case where $\sigma$ is admissible. In this paper, we show that such a bijection exists even if $\sigma$ is not admissible, possibly except some small rank cases., Comment: 48 PAGES

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., 25 pages, the final version, to appear in Nagoya Math. J

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 $\sigma : \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak g^{\sigma}$ be the fixed point subalgebra of $\mathfrak g$, and put $\underline U^-_q = U^-_q(\mathfrak g^{\sigma})$. Let $B$ be the canonical basis of $U_q^-$ and $\underline B$ the canonical basis of $\underline U_q^-$. $\sigma$ induces a natural action on $B$, and we denote by $B^{\sigma}$ the set of $\sigma$-fixed elements in $B$. Lusztig proved that there exists a canonical bijection $B^{\sigma} \simeq \underline B$ 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., Comment: 35 pages, final version, to appear in J. of Math. Soc. of Japan

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 $X_{\lambda}$ labelled by $r$-partitions $\lambda$ of $n$, and we consider the intersection cohomology $IC_{\lambda}$ associated to $X_{\lambda}$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $IC_{\lambda}$ 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 $r = 1, 2$., Comment: v2: 52 pages. Proposition 6.3 was revised, and some examples were added in section 7. Final version, to appear in Transformation Groups

This paper is a survey on a joint work with K. Sorlin concerning the theory of character sheaves on the exotic symmetric space. After explaining the historical background, we introduce character sheaves on this variety. By using those character sheaves, we show that modified Kostka polynomials, indexed by a pair of double partitions, can be interpreted in terms of the intersection cohomology of the orbits in the exotic nilpotent cone, as conjectured in Achar-Henderson.

In this paper, we construct a length function $n(w)$ for the complex reflection group $W = G(r, p, n)$ by making use of certain partitions of the root system associated to $\widetilde{W} = G(r, 1, n)$. We show that the function $n(w)$ yields the Poincaré polynomial $P_W(q)$. We give some characterization of this function in a way independent of the choice of the root system.

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., Comment: 28 pages, final version, to appear in J. of Algebra

Let $G = GL_N$ over an algebraically closed field of odd characteristic, and $\theta$ an involutive automorphism on $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1} \}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{uni}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{uni}$ 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 $G^{\iota\theta}_{uni}$ and irreducible representations of various symmetric groups., Comment: 76 pages. Final version, to appear in Tokyo J. of Mathematics

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by $r$-partitions $\lambda, \mu$ 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 $K_{\lambda,\mu}(t)$ by using an analogue of Lusztig's partition function, and showed that $K_{\lambda,\mu}(t)$ are polynomials in $t$ with non-negative integer coefficients. They conjecture that their $K_{\lambda,\mu}(t)$ coincide with $K^-_{\lambda,\mu}(t)$. In this paper, we show that their conjecture holds. We also discuss a multi-variable version of Kostka functions., Comment: 47 pages, v3: Final version, some references are added. To appear in SCIENCE CHINA Mathematics

Let H n , r \mathcal {H}_{n,r} be the Ariki-Koike algebra associated to the complex reflection group S n ⋉ ( Z / r Z ) n \mathfrak {S}_n\ltimes (\mathbb {Z}/r\mathbb {Z})^n , and let S ( Λ ) \mathcal {S}(\varLambda ) be the cyclotomic q q -Schur algebra associated to H n , r \mathcal {H}_{n,r} , introduced by Dipper, James and Mathas. For each p = ( r 1 , … , r g ) ∈ Z > 0 g \mathbf {p} = (r_1, \dots , r_g) \in \mathbb {Z}_{>0}^g such that r 1 + ⋯ + r g = r r_1 +\cdots + r_g = r , we define a subalgebra S p \mathcal {S}^{\mathbf {p}} of S ( Λ ) \mathcal {S}(\varLambda ) and its quotient algebra S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} . It is shown that S p \mathcal {S}^{\mathbf {p}} is a standardly based algebra and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of S ( Λ ) \mathcal {S}(\varLambda ) , which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic q q -Schur algebras associated to Ariki-Koike algebras H n i , r i \mathcal {H}_{n_i,r_i} of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} of type p \mathbf {p} , and prove the Schur-Weyl duality between H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} .

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ 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 $\mu$ is of the form $\mu = (-,\dots, -, \xi)$ and $\lambda$ is arbitrary, $K^-_{\lambda, \mu}(t)$ has a Lascoux-Sch\"utzenberger type combinatorial description., Comment: 21 pages

In this paper, we prove Lusztig’s conjecture for G F = S L n ( F q ) G^F = SL_n(\mathbf F_q) , i.e., we show that characteristic functions of character sheaves of G F G^F coincide with almost characters of G F G^F up to scalar constants, assuming that the characteristic of F q \mathbf F_q is not too small. We determine these scalars explicitly. Our result gives a method of computing irreducible characters of G F G^F .

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U q and on the tensor space

Let G G be a connected algebraic group defined over a finite field F q {\mathbf F}_q . For each irreducible character ρ \rho of G ( F q r ) G(\mathbf F_{q^r}) , we denote by m r ( ρ ) m_r(\rho ) the multiplicity of 1 G ( F q ) 1_{G({\mathbf F}_q)} in the restriction of ρ \rho to G ( F q ) G({\mathbf F}_q) . In the case where G G is reductive with connected center and is simple modulo center, Kawanaka determined m 2 ( ρ ) m_2(\rho ) for almost all cases, and then Lusztig gave a general formula for m 2 ( ρ ) m_2(\rho ) . In the case where the center of G G is not connected, such a result is not known. In this paper we determine m 2 ( ρ ) m_2(\rho ) , up to some minor ambiguity, in the case where G G is the special linear group. We also discuss, for any r ≥ 2 r \ge 2 , the relationship between m r ( ρ ) m_r(\rho ) with the theory of Shintani descent in the case where G G is a connected algebraic group.

Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall-Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture., Comment: 22 pages

Double Kostka polynomials are polynomials indexed by a pair of double partitions. As in the ordinary case, double Kostka polynomials are defined in terms of Schur functions and Hall-Littlewood functions associated to double partitions. In this paper, we study combinatorial properties of those double Kostka polynomials and Hall-Littlewood functions. In particular, we show that the Lascoux-Schutzenberger type formula holds for double Kostka polynomials in certain cases. Moreover, we show that the Hall bimodule introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions with two types of variables, which gives an alternate approach for their result., Comment: 33 pages, including tables of double Kostka polynomials

In our previous paper, Green functions associated to complex reflection groups G(e,1,n) were discussed. It involved a combinatorial approach to the Green functions of classical groups of type B_n or C_n. In this paper, we introduce Green functions associated to complex reflection groups G(e,p,n). We define Schur functions, Hall-Littlewood functions associated to G(e,p,n), and show that our Green functions are obtained as a transition matrix between those two kinds of symmetric functions. As a special case, it gives a combinatorial treatmnet of Green functions of type D_n., Comment: 33pages

Green functions of classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups G(e, 1, n) and study their combinatorial properties. We construct Hall–Littlewood functions and Schur functions in our scheme and show that such Green functions are obtained as a transition matrix between those two symmetric functions, as in the case of GLn.

Let g = glm1 ⊕ ··· ⊕ glmr be a Levi subalgebra of glm, with m = ∑ri = 1mi, and V the natural representation of the quantum group Uq(g). We construct a representation of the Ariki–Koike algebra Hn, r on the n-fold tensor space of V, commuting with the action of Uq(g), and prove the Schur–Weyl reciprocity for the actions of Uq(g) and Hn, r on it.

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function l(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space HW in terms of Demazure operators, and show that HW is isomorphic to the coinvariant algebra SW, which enables us to define a homogeneous basis on SW parametrized by w ϵ W.

Letp be an odd prime and\(\mathbb{F}_p \) the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space\(\mathbb{F}_p^n \) and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field\(\mathbb{Q}(\sqrt p )\) (wherep≡1 (mod 4)) over the field ℚ of rational numbers can be expressed by means of these formulas.

In this paper, the Shintani descent of irreducible characters of finite special linear groups is discussed. We parametrize irreducible characters of SL n ( F q ) by making use of (modified) generalized Gelfand–Graev characters. Shintani descent of irreducible characters of SL n ( F q m ) is determined by investigating the Shintani descent of generalized Gelfand–Graev characters. Our results hold for arbitrary p and q .

Let V be an 2n-dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V), and H = Sp(V) the symplectic group contained in G. For a positive integer r > 1, we conisder the variety X = G/H \times V^{r-1}, on which H acts diagonally. X is called the exotic symmetric space of level r. Let W_{n,r} be the complex reflection group G(r,1,n). In this paper, generalizing the result for the case where r = 2, we show that there exists a natural bijective correspondence (Springer correspondence) between the set of irreducible representations of W_{n,r} and a certain set of H-equivariant simple perverse sheaves on X_{uni}$, where X_{uni} is the "unipotent part" of X. We also consider a similar problem for G \times V^{r-1}, where G = GL(V) for a finite dimensional vector space V., 69 pages

Let V be a 2n dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V) and H = Sp_{2n} be the symplectic group contained in G. We call X = G/H \times V the exotic symmetric space, since its "unipotent" part is isomorphic to Kato's exotic nilcone. Let W be the Weyl group of type C_n. Kato established the Springer correspondence between irreducible characters of W and H-orbits in the exotic nilcone, based on the Ginzburg theory of affine Hecke algebras. In this paper, we develpoe a theory of character sheaves on X, and give an alternate proof for the Springer correspondence based on the theroy of character sheaves., 56 pages, the proof of Proposition 4.8 revised, and typos corrected. The results unchanged

Let G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula., 24 pages

In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In this paper, we consider a similar problem for the case of classical groups Sp_{2n}, SO_N with arbitrary characteristic. Our result enables us, in particular, to compute Green functions in bad characteristic case, which was not known before., Comment: 31 pages

Let G be a simple algebraic group over C with the Weyl group W. For a unipotent element u of G, let B_u be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup W_L. Assume that u is in L and let B_u^L be the corresponding variety for L. The induction theorem of Springer representations describes the W-module structure of H*(B_u) in terms of the W_L-module structure of H*(B_u^L). In this paper, we prove a certain refinement of the induction theorem by considering the action of a cyclic group of order e on H^*(B_u). As a corollary, we obtain a description of the values of Green functions at e-th root of unity., 18 pages. In this revised version, some errors are corrected. In particular some restrictions are added to the main results

Hall-Littlewood functions and Green functions associated to complex reflection groups $W = G(r, 1, n)$ were constructed in [S1] by means of symbols, which are a generalization of partitions. In this paper, we consider such functions in the case where the symbols are of very special type, called "limit symbols". The situation becomes simple, and is close to the case of symmetric groups when the symbols tends to the "limit". In the case where $W$ is a Weyl group of type $B_n$, we give a closed formula for Hall-Littlewood functions, and verify some of the conjectures stated in [S1] for the case of Green functions attached to limit symbols.

Let G be a connected, reductive algebraic group defined over a finite field, with Frobenius endomorphism F: G → G. In a previous article by the first author [3], the irreducible representations, in an algebraically closed field K of characteristic zero, of the Hecke algebra Ή of a Gelfand-Graev representation γ of the finite reductive group G F were determined. They are parametrized by pairs (T, θ), with T an F-stable maximal torus of G, and θ an irreducible character of the finite torus T F . Each irreducible representation f T, θ can be factored, \({{f}_{{T,{\kern 1pt} \theta }}}{\mkern 1mu} = {\mkern 1mu} \tilde{\theta } {\mkern 1mu} ^\circ {{f}_{T}} \), with f T a homomorphism of algebras, independent of θ, from H to the group algebra KTF, and \({\tilde{\theta }} \) the extension of θ to an irreducible representation of the group algebra of the torus T F .

(1982). On the green polynomials of a chevalley group of type f4. Communications in Algebra: Vol. 10, No. 5, pp. 505-543.