Fernando Szechtman
We show that in characteristic 2, the Steinberg representation of the symplectic group Sp(2n,q), q a power of an odd prime p, has two irreducible constituents lying just above the socle that are isomorphic to the two Weil modules of degree (q^n-1)/2.
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The modular reduction of the Steinberg lattice of the general linear group is studied
We study the Steinberg lattice of the general linear group when reduced modulo a prime different from the defining characteristic.
Let $G$ be a finite group of Lie type and $\St_k$ be the Steinberg representation of $G$, defined over a field $k$. We are interested in the case where $k$ has prime characteristic~$\ell$ and $\St_k$ is reducible. Tinberg has shown that the socle of $\St_k$ is always simple. We give a new proof of this result in terms of the Hecke algebra of $G$ with respect to a Borel subgroup and show how to identify the simple socle of $\St_k$ among the principal series representations of~$G$. Furthermore, we determine the composition length of $\St_k$ when $G=\GL_n(q)$ or $G$ is a finite classical group an…
We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an…
We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.
A conjecture on the tensor ideal for an elementary p-group generated by the restriction of a Steinberg module
本文针对初等阿贝尔p-群的模表示与最大非投射受限Steinberg SL_2-模的张量积是否为受限倾斜模这一猜想，提供了支持性证据，并指出其成立将对正特征下不可压缩对称张量范畴的分类产生重要影响。
The Steinberg Tensor Product Theorem for General Linear Group Schemes in the Verlinde Category
本文将约化代数群的Steinberg张量积定理推广至Verlinde范畴$\mathrm{Ver}_p$中任意对象$X$上的一般线性群概形$GL(X)$，解决了其模表示的特征标问题。
We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural embeddings of the classical groups, inductive systems of representations with totally bounded weight multiplicities are classified. An analogue of the Steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved.
A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where a key combinatorial ingredient comes from the Mullineux conjecture on modular representations of the symmetric group. A Steinberg tensor product theorem for the ortho-symplectic supergroup is also obtained.
Presenting restricted Steinberg modules of general linear groups
研究给出域上斯坦伯格模各种局部限制的生成元和表示，将其与数域中局部对称空间的部分博雷尔 - 塞尔紧化相关联，还厘清相关表示关系，并在附录中关联表示存在性与斯坦伯格VB - 代数的科祖尔性质。
Flag Space, Matroidal Schur Algebras and the Steinberg Representation
研究通过生成元和关系定义拟阵舒尔代数的扩展版本，在射影几何中关联其与\(GL(n,q)\)模斯坦伯格表示结构，一般情形下证明其有三角或里迪分解，还得到原始代数新描述并阐明与旗空间联系。
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …
In this paper branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group $Sp_{2n}(K)$ to the naturally embedded subgroup $Sp_{2n-2}(K)$ is determined. As a corollary, inductive systems of fundamental representations for $Sp_{\infty}(K)$ are classified. The submodule structure of the fundamental Weyl modules is refined.
Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. An…
In this paper we study highest weight and standardly stratified structures on modular analogs of categories $\mathcal{O}$ over quantizations of symplectic resolutions and show how to recover the usual categories $\mathcal{O}$ (reduced mod $p\gg 0$) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of $\mathbb{Z}$ and is equipped with a Hamiltonian action of a torus $T$ that has finitely many fixed points. We consider algebras $\mathcal{A}_λ$ of global sections of a quantization in characterstic $p\gg 0$, where $λ…
Symplectic model for ladder and unitary representations
本文研究特征零非阿基米德局部域上四元数可除代数对应的$GL_n(D)$的辛模型问题，分类了具辛模型的阶梯表示，验证了Steinberg表示满足Prasad猜想，证明了诱导表示辛模型的遗传性及Prasad猜想的部分内容。
We use the twisted topological trace formula developed in an earlier paper to understand liftings from symplectic to general linear groups. We analyse the lift from $\SP_{2g}$ to $\PGL_{2g+1}$ over the ground field $\Q$ in further detail, and we get a description of the image of this lift for the $L^2$ cohomology of $\SP_{2g}$ (which is related to the intersection cohomology of the Shimura variety attached to $\GSp_{2g}$) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups. This description may be used to un…
In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Tay…
The purpose of the paper is to derive linkage principle for modular representations of ortho-symplectic supergroups. We follow the approach of Doty and investigate in detail the representation theory of the orthosymplectic group $OSP(2|1)$ and that of its Frobenius thickening. Using the description of flags and adjacent Borel supersubgroups we derive first the strong linkage for the Frobenius thickening $G_rT$ of the orthosymplectic supergroup $G$ of type $SpO(2m|2n+1)$ and $SpO(2m|2n)$. Based on this, we derive the linkage principle for orthosymplectic supergroup $SpO(2m|2n+1)$ and $SpO(2m|2n…
We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow with the dimensions of the irreducible Sp(2g,F_p)-modules that appear in the first approximation to representations of mapping class groups of surfaces in Integral Topological Quantum Field Theory. For this purpose, we derive a trigonometric formula for the dimensions of Gow's representations. This formula is equivalent to a special case of a formula contained in unpublished work of Foulle. Our direct proof is simpler than the proof of Foulle's more general result, and is modeled…
Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result showing that a mod $\ell$ Galois representation arising from a stable cuspidal automorphic representation of the unitary similitude group $G=\mathrm{GU}(1,2)$ which is Steinberg at an inert prime $p$ can also arise from an automorphic representation of $G$ that is unramified at $p$.
The $\ell$-modular local theta correspondence
本文在模表示理论语境下，证明了非阿基米德局部域上 $p$-进域的辛-正交与酉-酉对偶对在系数域特征足够大时存在双射对应，即模局部theta对应。
We describe an essential improvement of our recent algorithm for computing cohomology of Lie (super)algebra based on partition of the whole cochain complex into minimal subcomplexes. We replace the arithmetic of rational numbers or integers by a much cheaper arithmetic of a modular field and use the inequality between the dimensions of cohomology H over any modular field F_p = Z/pZ and over Q: dim H(F_p) >= dim H(Q). With this inequality we can, by computing over arbitrary F_p, quickly find the (usually, rare) subcomplexes for which dim H(F_p) > 0 and then carry out the full computation over Q…
On two modular geometric realizations of an affine Hecke algebra
本文通过构造三类范畴的幺半等价，将正特征域上仿射 Hecke 代数与 Harish-Chandra 双模、Steinberg 簇等变凝聚层及仿射旗簇构造层联系起来，并据此证明了 Finkelberg-Mirković 关于主块几何实现的猜想。
An analogue of Whittaker reduction for group-valued moment maps
本文为半单复Poisson-Lie群的Poisson作用构造了Whittaker约化的类似，该约化沿幂零轨道横截面进行，并在Dirac几何框架下描述其辛叶。
Let K be an algebraically closed field of characteristic p>0 and let Sp(2m) be the symplectic group of rank m over K. The main theorem of this article gives the character of the rational simple Sp(2m)-modules with fundamental highest weight as an explicit alternating sum of characters of Weyl modules. One obtains several formulae for the dimensions of these simple modules, what allows us to investigate the asymptotic behavior of these dimensions, for a given p, when the rank m is growing towards infinity. One also gets the simple Weyl modules with fundamental highest weight, and the article en…
Tempered Symplectic Distinction For Quaternionic $GL_n(D)$: A Conjecture of PRASAD
本文证明了 Prasad 关于四元数一般线性群上温和表示辛区分的修正猜想，给出了可区分温和表示的精确刻画。
We complete the l-modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups SO_7(q) and the symplectic groups Sp_6(q) for all prime powers q and all odd primes l different from the defining characteristic.
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. We generalize the Kudla-Millson relation between intersection numbers of cycles and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the generating series of the cycles give rise to vector-valued Siegel modular forms. The underlying correspondence between the highest weights of the orthogonal and the symplectic group coincides with the one obtained by Adams for which we provide a geometric interpretation.
In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the "correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has been made in a number of different directions, most notably by Braue…
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。
Modular reduction of complex representations of finite reductive groups
本文研究有限李型群的单幂表示的Brauer-Nesbitt降，通过Weyl模的限制表达，部分验证了Lusztig的猜想，引入了新的基并探讨了导出范畴中的对象及几何解释。
We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum…
For p>3 a prime, and g>2 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of L_p(lambda) for these highest weights.
We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This provides an example of modularity, explicit uniformization and hyperbolic uniformization of arithmetic type for a higher genus algebraic space curve. In particular, it gives a new example for Hilbert's 22nd problem. This gives $21$ modular equations of order $13$, which greatly improve the result of Ramanujan and Evans on the construction of modular equations …
We construct via generators and relations, generalized Weil representations for analogues of classical $SL(2,k), k$ a field, over involutive base rings $(A, \ast).$ This family of groups covers different kinds of groups, classical and non classical. We give some examples that include symplectic groups as well as non classical groups like $SL_\ast(2,A_m), $ where $A_m$ is the finite modular analogue of the algebra of real m-jets in one dimension with its canonical involutive symmetry.
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