Jacquet modules of principal series generated by the trivial $K$-type

Muller type irreducibility criterion for generalized principal series

We obtain an irreducibility criterion for generalized principal series, extending known and frequently employed results for principal series. Our approach rests on a newly observed semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the Weyl group of a general Levi subgroup is not a Coxeter group. Our statements on irreducibility are formulated in terms of a subgroup of the Knapp--Stein R-group, which arises naturally from our decomposition of the Weyl…

Invariant triple functionals over $U_q\frak{sl}_2$

We define a $q$-deformation of Jacquet-Langlands principal series representations of $GL_2(R)$ and prove the uniqueness of an invariant triple functional on them using the method of H.Y.Loke.

关于由Whittaker模诱导的一些非主局部解析表示

On some non-principal locally analytic representations induced by Whittaker modules

本文研究由连通分裂约化p进李群的Whittaker模诱导的非主局部解析表示，证明其不可容许且拓扑不可约、朴素Jacquet函子对所有抛物子群消失，且不满足Kohlhaase的超尖点定义。

与极大抛物子群相关的扭Jacquet模及$GL(n, q)$的尖表示

Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$

本文研究有限域上一般线性群尖表示关于极大抛物子群的扭Jacquet模的结构，通过建立扭Jacquet函子的范畴等价并构造具有泛性质的表示，直接解决了所有参数情形下的模结构问题。

实约化群Whittaker积分的均匀缓增性

Uniform temperedness of Whittaker integrals for a real reductive group

本文研究实约化群广义主序列中Whittaker向量及其Jacquet积分，通过函数方程建立均匀估计，并证明Harish-Chandra型Whittaker积分的均匀缓增性，进而得到Whittaker Schwartz函数空间上傅里叶变换的快速衰减。

Locally trivial principal homogeneous spaces and conjugacy theorems for Lie algebras

For g a split semisimple Lie algebra over a field of characteristic 0, we link locally trivial principal homogeneous spaces over Spec(R) to the question of conjugacy of maximal abelian diagonalizable subalgebras of g(R).

Potential level-lowering for GSp(4)

In this preprint, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place w, and forms with a tamely ramified principal series at w. Thus, after base change to a totally real finite solvable extension, one can often lower the level at w. For the proof, we first establish an analogue of the Jacquet-Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity mod cent…

The double cover of odd general spin groups, small representations and applications

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series, it is an automorphic representation and decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of SO(2n+1) of Bump, Fr…

Triviality properties of principal bundles on singular curves-II

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the si…

一个深度为零的主序列块，其赫克代数具有非平凡的二上循环

A depth-zero principal-series block whose Hecke algebra has a non-trivial two-cocycle

本文证明文献中一个被提议的反例并非Kim-Yu类型赫克代数同构定理的反例，并展示该深度为零主序列类型的赫克代数具有非平凡二上循环的有趣性质。

Triviality properties of principal bundles on singular curves

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $π_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve.

Branching of unitary $\operatorname{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology

Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $Π$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $Π$ occurs inside a principal series representation of $G$, induced from the $\operatorname{O}(n)$-representation $\bigwedge\nolimits^p(\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to…

Centres of blocks of finite groups with trivial intersection Sylow $p$-subgroups

For finite groups $G$ with non-abelian, trivial intersection Sylow $p$-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of $G$ and the centre of the Brauer correspondent. This was already known for the Suzuki groups; the result will be generalised to cover more groups with trivial intersection Sylow $p$-subgroups.

有限群的国内表示类型块中的平凡源特征

Trivial source characters in blocks of domestic representation type

研究国内表示类型块中平凡源模块的普通特征，确定其在不同块情况下的表现。

Singularities of Intertwining Operators and Decompositions of Principal Series Representations

In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^Gλ$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $χ_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connec…

等价于主块上一个条件的群论条件

A group-theoretic condition equivalent to a condition on principal blocks

本文提出一个群论条件，等价于有限群在指定素数集的主块中仅含平凡复不可约特征的性质。

Principal Schottky Bundles over Riemann surfaces

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G…

不同素数主块中的特征标次数

Character degrees in principal blocks for distinct primes

本文证明阶被不同素数整除的有限群在特定条件下存在同时位于主块且次数不被素数整除的非平凡不可约特征标，推广了前人结果并完成相关猜想的证明。

Projective indecomposable modules and quivers for monoid algebras

We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, $\mathscr J$-trivial and $\mathscr R$-trivial monoids and left regular bands of groups.