Higher-rank zeta functions andSLn-zeta functions for curves

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

Torus quotient of Richardson varieties in orthogonal and symplectic grassmannians

For any simple, simply connected algebraic group [Formula: see text] of type [Formula: see text] and [Formula: see text] and for any maximal parabolic subgroup [Formula: see text] of [Formula: see text], we provide a criterion for a Richardson variety in [Formula: see text] to admit semistable points for the action of a maximal torus [Formula: see text] with respect to an ample line bundle on [Formula: see text].

Towards a Goldberg–Shahidi pairing for classical groups

Let{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical

Let P = M N be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group G over a p-adic field F. Assume that there exists w0 ∊ G(F) that normalizes M and conjugates P to an opposite parabolic subgroup. When N has a Zariski dense Int M-orbit, F. Shahidi and X. Yu described a certain distribution D on M(F), such that, for irreducible unitary supercuspidal representations π of M(F) with is irreducible if and only if D( f )≠ 0 for some pseudocoefficient f of π. Since this irreducibility is conjecturally related to π arising via transfer from certain twisted endoscopic groups of M, it is of interest to realize D as endoscopic transfer from a simpler distribution on a twisted endoscopic group H of M. This has been done in many situations where N is abelian. Here we handle the standard examples in cases where N is nonabelian but admit a Zariski dense Int M-orbit.

Geometric classification of semidirect products in the maximal parabolic subgroup of Sp(2, ℝ)

We classify up to conjugation by [Formula: see text] (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of [Formula: see text] by means of an essentially geometric argument. This classification has already been established in [G. S. Alberti, L. Balletti, F. De Mari and E. De Vito, Reproducing subgroups of [Formula: see text]. Part I: Algebraic classification, J. Fourier Anal. Appl. 9(4) (2013) 651–682] without geometry, under a stricter notion of equivalence, namely, conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.

Restricting unipotent characters in special orthogonal groups

For all prime powers $q$ we restrict the unipotent characters of the special orthogonal groups $\text{SO}_{5}(q)$ and $\text{SO}_{7}(q)$ to a maximal parabolic subgroup. We determine all irreducible constituents of these restrictions for $\text{SO}_{5}(q)$ and a large part of the irreducible constituents for $\text{SO}_{7}(q)$.

Branching laws for small unitary representations of GL(n, ℂ)

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2r) case

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO −(2n, 2r). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O −(2n, 2r).

COMPUTATION OF A TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(3, E)

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let [Formula: see text] be the involution of E over F. The representation π of GL (3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution [Formula: see text]. We compute — by purely local means — the σ-twisted character [Formula: see text] of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U (2, E/F) of GL (3, E). In particular π is σ-elliptic, that is, [Formula: see text] is not identically zero on the σ-elliptic set.