The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: Part II

AbstractWe study the Rankin–Selberg integral for a pair of representations of ${\rm SO}_{2l}\times {\rm GL}_{n}$, where ${\rm SO}_{2l}$ is defined over a local non-Archimedean field and is either split or quasi-split. The integrals span a fractional ideal, and its unique generator, which contains any pole which appears in the integrals, is called the greatest common divisor (gcd) of the integrals. We describe the properties of the gcd and establish upper and lower bounds for the poles. In the tempered case we can relate it to the $L$-function of the representations defined by Shahidi. Results of this work may lead to a gcd definition for the $L$-function.

Download Full-text

Low-dimensional representations of finite orthogonal groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000037 ◽

2021 ◽

pp. 1-22

Author(s):

KAY MAGAARD ◽

GUNTER MALLE

Keyword(s):

Orthogonal Groups ◽

Brauer Characters ◽

Low Dimensional

Abstract We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

Download Full-text

STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000177 ◽

2021 ◽

pp. 1-48

Author(s):

Federico Scavia

Keyword(s):

Finite Groups ◽

Reductive Groups ◽

Orthogonal Groups ◽

De Rham Cohomology ◽

Algebraic Stacks ◽

Steenrod Operations ◽

De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

Download Full-text

Higher pullbacks of modular forms on orthogonal groups

Forum Mathematicum ◽

10.1515/forum-2020-0066 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Brandon Williams

Keyword(s):

Modular Forms ◽

Differential Operators ◽

Jacobi Form ◽

Siegel Modular Forms ◽

Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

Download Full-text