Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$, $q>3$ odd via presentation and compatibility of methods

模Weil表示与自守形式在同余下的相容性

Modular Weil representation and compatibility of cuspidals with congruences

本文研究了在ℓ-模设定下Weil表示的推广，并探讨了自守形式在同余下的相容性问题。

Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

We construct a complex linear Weil representation $ρ$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $ρ…

Geometric Weil representations for star-analogues of SL(2,k)

We present here an elementary geometric approach to the construction of Weil representations of the star-analogues $SL_\ast(2,A), A$ a ring or algebra with involution $\ast$, of the group SL(2, k), k a field, reminiscent of the quantum groups $SL_q(2,A)$. We review as well the elementary construction of Weil representations for these groups via generators and relations, which uses the Bruhat presentation available in many cases. We compare the representations obtained by both methods in the non - classical case of the finite truncated polynomial algebra $A_m$ of degree $m$ with its canonical i…

容直和$ q $-字符在仿射Kac-Moody量子对称对中的兼容性：准分裂情况

Compatibility of Drinfeld presentations and $q$-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case

本文研究了准分裂仿形Kac-Moody量子对称对中Drinfeld级数的因子分解和余乘公式，并构造了与Frenkel和Reshetikhin的$ q $-字符同态相容的边界类比。

The categorical Weil representation

In a previous work the authors gave a conceptual explanation for the linearity of the Weil representation over a finite field k of odd characteristic: There exists a canonical system of intertwining operators between the Lagrangian models of the Heisenberg representation. This defines a canonical vector space H(V) associated with a symplectic vector space V over k. In this paper we prove a general theorem about idempotents in categories, and we use it to solve the sign problem, formulated by Bernstein and Deligne, on the compatibility between the associativity constraint and the convolution st…

分裂仿射Kac-Moody量子对称对的Drinfeld表示的相容性

Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs

本文证明分裂仿射Kac-Moody量子对称对中Drinfeld-Cartan算子的分解与余积公式，并证明边界$q$-特征映射与通常$q$-特征映射相容，同时给出扩展仿射Weyl群基本权的显式约化表达式。