Yingjue Fang, Binyong Sun
We formulate and prove Chevalley's theorem in the setting of affine Nash groups. As a consequence, we show that the semi-direct product of two almost linear Nash groups is still an almost linear Nash group.
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A toroidal affine Nash group is the affine Nash group analogue of an anti-affine algebraic group. In this note, we prove analogues of Rosenlicht's structure and decomposition theorems: (1) Every affine Nash group $G$ has a smallest normal affine Nash subgroup $H$ such that $G/H$ is an almost linear affine Nash group, and this $H$ is toroidal. (2) If $G$ is a connected affine Nash group, then there exist a largest toroidal affine Nash subgroup $\ant{G}$ and a largest connected, normal, almost linear affine Nash subgroup $\aff{G}$. Moreover, we have $G=\ant{G}\aff{G}$, and $\ant{G}\cap \aff{G}$ …
The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of cha…
We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type $C$. These identities express the product $e^μ \, \mathrm{gch} \, V_{x}^{-}(λ)$ of the (one-dimensional) character $e^μ$, where $μ$ is a (not necessarily dominant) minuscule weight, with the graded character $\mathrm{gch} \, V_{x}^{-}(λ)$ of the level-zero Demazure submodule $V_{x}^{-}(λ)$ over the quantum affine algebra $U_{\mathsf{q}}(\mathfrak{g}_{\mathrm{af}})$ as an explicit finite linear combination of the graded char…
By the algebraization of affine Nash groups, a connected affine Nash group is an abelian Nash manifold if and only if its algebraization is a real abelian variety. We first classify real abelian varieties up to isomorphisms. Then with a bit more efforts, we classify abelian Nash manifolds up to Nash equivalences.
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to …
Locally Semialgebraic Superspaces and Nash Supermanifolds
本文提出局部半代数超空间与纳什超流形理论，利用 Batchelor 定理证明其可由向量丛导出，并证明仿射纳什流形上拟凝聚层的高阶上同调消失。
Characters for extended affine Lie algebras; a combinatorial approach
本文用组合方法研究扩展仿射李代数的特征标与嘉当自同构，证明几乎所有此类代数中有限阶嘉当自同构都是对角的，且对应组合映射为特征标。
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) qua…
Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these…
How many generators and relations does ${\mathrm SL}_n({\mathbb F}_q[t, t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac-Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac-Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a conseque…
For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.
Generic spherical unitary dual for Chevalley groups over local fields
本文研究局部域上Chevalley群的一般球面酉对偶，通过结合多种代数工具证明其与域无关，且具有统一的简单描述。
We show that every connected affine algebraic supergroup defined over a field k, with diagonalizable maximal torus and whose tangent Lie superalgebra is a k-form of a complex simple Lie superalgebra of classical type is a Chevalley supergroup, as it is defined and constructed explicitly in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014].
Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
本文基于Lenart和Postnikov的λ链，推导了广义旗 manifold 中 Schubert 胞 motivic Chern 类与线丛类相乘的 Chevalley 公式，并探讨了稳定包在 T*(G/B) 中的墙穿越公式及多项式系数的正性性质。
We develop a general theory of algebraic group superschemes, which are not necessarily affine. Our key result is a category equivalence between those group superschemes and Harish-Chandra pairs, which generalizes the result known for affine algebraic group superschemes. Then we present the applications, including the Barsotti-Chevalley Theorem in the super context, and an explicit construction of the quotient superscheme $\mathbb{G}/\mathbb{H}$ of an algebraic group superscheme $\mathbb{G}$ by a group super-subscheme $\mathbb{H}$.
Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a $\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-th…
Let $\mathfrak{g}(A)$ be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fix-point Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of $\mathfrak{k}(A)$ in terms of a Chevalley…
It is shown that there is an order isomorphism $ϕ'$ from the poset $V$ of $B\times B$-orbits on the wonderful compactification of a semi-simple adjoint group $G$ with Weyl group $W$ to an interval in reverse Chevalley-Bruhat order on a non-canonically associated Coxeter group $\hat{W}$ (in general neither finite nor affine). Moreover, $ϕ'$ preserves the corresponding Kazhdan-Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan-Lusztig polynomials for the analogues of $V$ for general Coxeter groups $W$ is completed by reducing it by a similar order isomorphism to known r…
Divided Power Integral forms of Affine Algebras
本文证明仿射Kac-Moody代数的Drinfeld生成元除幂生成积分形式，并与Chevalley生成元所得积分形式比较，发现扭A型外一致、扭A型中更小，且高秩时虚向量子代数非多项式，遂在低秩情形引入两种新积分形式以获多项式性。
Let V denote a finite dimensional vector space over an algebraically closed field. Let U_0, U_1,..., U_d denote a sequence of nonzero subspaces whose direct sum is V. Let R:V \to V and L:V \to V denote linear maps with the following properties: for 0 \leq i \leq d, R U_i \subseteq U_{i+1} and L U_i \subseteq U_{i-1} where U_{-1}=0, U_{d+1}=0; for 0 \leq i \leq d/2, the restrictions R^{d-2i}: U_i \to U_{d-i} and L^{d-2i}: U_{d-i} \to U_i are bijections; the maps R and L satisfy the cubic q-Serre relations where q is nonzero and not a root of unity. Let K:V \to V be the linear map such that (K-q…
Braid group actions, Baxter polynomials, and affine quantum groups
本文通过修正经典构造，将辫群作用提升到Yangian交换子代数，并通过对偶与量子环代数联系，证明了循环性条件的等价性。
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