Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n-1) for principal series representations

Let k be a local field.Let Iνand I_ν′be smooth principal series representations of GL_n(k) and GL_n-1(k),respectively.The Rankin-Selberg integrals yield a continuous bilinear map I_ν×I_ν′→C with a certain invariance property.We study integrals over a certain open orbit that also yield a continuous bilinear map II_ν×I_ν′→C with the same invariance property and show that these integrals equal the Rankin-Selberg integrals up to a...     

On representations of GL(n) distinguished by GL(n − 1) over a p-adic field

For a nonarchimedean local field F, let GL(n):= GL(n, F) and GL(n?1) be embedded in GL(n) via g ? ( _{0 1} ^{g 0} ). Let π be an irreducible admissible representation of GL(n) for n ≥ 3. We prove that π is GL(n ? 1)-distinguished if and only if the Langlands parameter L(π) associated to π by the Local Langlands Correspondence has a subrepresentation L(11 _n?2) of dimension n?2 corresponding to the trivial representation of GL(n?2) such that the two-dimensional quotient L(π)/L(11 _n?2) corresponds either to an infinite-dimensional representation or the one-dimensional representations $\nu ^{ \pm (\tfrac{{n - 2}}{2})} $ of GL(2). We also prove that, for a parabolic subgroup P of GL(n) and an irreducible admissible representation ρ of the Levi subgroup of P, $\dim _\mathbb{C} (Hom_{GL(n - 1)} [ind_P^{GL(n)} (\rho ),\mathbb{I}_{n - 1} ]) \leqslant 2$ . For the standard Borel subgroup B _n of GL(n) and characters x _i of GL(1), we classify all representations ξ of the form $ind_{B_n }^{GL(n)} (\chi _1 \otimes \cdots \otimes \chi _n )$ for which $\dim _\mathbb{C} (Hom_{GL(n - 1)} [\xi ,\mathbb{I}_{n - 1} ]) = 2$ .     

Tensor invariants of the Lie algebra sl2(ℂ[t]) and fundamental representations of the Lie algebras\hat p_{2n}

Generators of the space of tensor invariants of the Lie algebra Sl₂([t]) are constructed. It is proved that the restrictions of a spinor representation of the affine Lie algebra to and form a dual pair. A realization of the fundamental representations of the Lie algebra is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 137–144, 1989.     

Period Integrals and Rankin–Selberg L-functions on GL(n)

We compute the second moment of a certain family of Rankin–Selberg L-functions L(f ×?g, 1/2) where f and g are Hecke–Maass cusp forms on GL(n). Our bound is as strong as the Lindel?f hypothesis on average, and recovers individually the convexity bound. This result is new even in the classical case n?=?2.     

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Mathematische Zeitschrift - Let $$\pi $$ be an irreducible cuspidal representation of $$\mathrm {GL}_{kn}(\mathbb {F}_q)$$ . Assume that $$\pi = \pi _{\theta }$$ , corresponds to a regular...     

Direct Sums of Mod p Characters of Gal and the Homology of GL(n, ℤ)

The paper classifies multilincnr algebras, known as comlrans algebras, that are determined by pairs of matrices and have dimensions which are less than or equal to two. The comtrans algebras are isomophic if and only if the corresponding pairs of matrices arc multiplicatively cogredient by a matrix M.     

Correction to: On positive integers n with σl(2n + 1) < σl(2n)

Periodica Mathematica Hungarica -     

On the Size of a Cap in PG(n,q) with q Even and n ≥ 3

Let m2(n,q), m2(n,q) be, respectively, the maximum value, the second largest value of k for which there exists a complete k-cap in PG(n,q). In this paper, the known upper bound on m2(3,q), q even, q 8, is improved. This new upper bound on m2(3,q) is then used to improve the upper bounds on m2(n,q), q even, q 8 and n 4.     

Infinite-Dimensional Representations of the Lie Algebra \mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right) Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL\left( {n,\mathbb{C}} \right)

Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z ^±, where Z ⁺ and Z ^? are the triangular unipotent subgroups. Representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.     

Isomorphism of the Yangian Yħ(A(m,n)) of the Special Linear Lie Superalgebra and the Quantum Loop Superalgebra Uħ(LA(m,n))

Theoretical and Mathematical Physics - Using the approach of Gautam and Toledano Laredo, we construct an explicit isomorphism of the Yangian Y?(A(m, n)) of the special linear Lie superalgebra...     

A new class of planar π-spaces and some related topics: (n,d)-systems and (σ,n)-spaces

Summary Planar spaces with planes isomorphic to PG(d, Q) or to AG(d, Q), with d 3, are presented, and a natural generalization of -spaces, namely the (, n)-spaces, is also studied. For this purpose, we use the language of (n, d)-systems, which was introduced and studied by G. Tallini [13], and for which we give a brief sketch of the theory.Dedicated to Professor M. Barner on the occasion of his 65^th birthday