arXiv subjects

Ehud Moshe Baruch

Publications and source records attributed to Ehud Moshe Baruch.

11 recordsLinked to original sources

arXiv2026

Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of $\SL_2(\mathbb Q_p)$

We determine an explicit compact Hecke subalgebra for the metaplectic double cover of $\SL_2(\mathbb Q_p)$ at the congruence subgroup $K_0(p^n)$, for odd $p$, and use it to study local newforms of prescribed quadratic type. We describe the supporting double cosets, generators, relations, characters, and corresponding $\ov K$-types, and compute the action of the resulting Hecke operators on $(K_0(p^n),\eta)$-isotypic vectors in principal series, Weil, Steinberg, and supercuspidal representations. Thus the conductor relevant throughout is the $\eta$-conductor, rather than the minimum over all characters. In the supercuspidal case, the metaplectic calculation is reduced to the corresponding linear strongly cuspidal type, making explicit the distinction between unramified and ramified $L$-packets. We also compare the operator $W_{m-1}$ with Ishimoto's local realization of Ueda's twisting operator: after fixed-level compression and a lifted $\GL_2$-conjugation, the two actions agree up to an explicit scalar and a parity-dependent change of type. These results provide the local odd-prime counterpart to the Hecke-algebra methods used in the theory of half-integral-weight newforms and minus spaces.

math.NT↗

arXiv2024

Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field

Let $G=GL_{n}(F)$ and let $(π_{St},V)$ be a (generalized) Steinberg representation of $G$. It is well known that the space of Iwahori fixed vectors in $V$ is one dimensional. The Iwahori Hecke algebra acts on this space via a character. We determine the value of this character on a particular Hecke algebra element and use this action to determine in full the Whittaker function associated with an Iwahori fixed vector generalizing a result of Baruch and Purkait for $GL_2(F)$. We show that the Iwahori fixed vector is "new" in the sense that it is not fixed by any larger parahoric. We also show that the restriction of the (generalized) Steinberg representation to $SL_n(F)$ remains irreducible hence we get the Whittaker function attached to a Steinberg representation of $SL_n(F)$.

math.RT↗

arXiv2024

Iwahori Spherical Whittaker Functions for Steinberg Representations

Let $G(F)$ be a split reductive group over a $p$-adic field $F$ and let $(\pi_{St},V)$ be a (generalized) Steinberg representation of $G(F)$. It is known that the space of Iwahori fixed vectors in $V$ is one dimensional. The Iwahori Hecke algebra acts on this space via a character. We determine this fixed vector and use the Hecke algebra action on it to determine in full the Whittaker function associated with this Iwahori fixed vector. This generalizes our previous result for $GL_n(F)$.

math.RT↗

arXiv2018

Strong contraction, the mirabolic group and the Kirillov conjecture

We lift any (infinitesimal) unitary irreducible representation of $GL_n(\mathbb{R})$ to a family of representations that strongly contracts to a certain type of (infinitesimal) unitary irreducible representations of $\mathbb{R}^n\rtimes {M}_n$, with $M_n$ being the mirabolic subgroup of $GL_n(\mathbb{R})$. For the case of $n=2$ we obtain the full unitary dual of $\mathbb{R}^2\rtimes {M}_2$ as a strong contraction. We demonstrate the role of the Kirillov conjecture and Kirillov model for these contractions.

math-ph↗

arXiv2018

Newforms of half-integral weight: the minus space of S_{k+1/2}(Γ_0(8M))

We compute generators and relations for a certain $2$-adic Hecke algebra of level $8$ associated with the double cover of $\mathrm{SL}_2$ and a $2$-adic Hecke algebra of level $4$ associated with $\mathrm{PGL}_2$. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the $2$-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight $k+1/2$ and level $8M$ where $M$ is odd and square-free. Using these operators and our previous results on half-integral weight forms of level $4M$ we define a subspace of the space of half-integral weight forms as a common $-1$ eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace which we call the minus space is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight $2k$ and level $4M$. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

math.NT↗

arXiv2017

Gabor analysis as contraction of wavelets analysis

We use the method of group contractions to relate wavelets analysis and Gabor analysis. Wavelets analysis is associated with unitary irreducible representations of the affine group while Gabor analysis is associated with unitary irreducible representations of the Heisenberg group. We obtain unitary irreducible representations of the Heisenberg group as contractions of representations of the extended affine group. Furthermore, we use these contractions to relate the two analyses, namely we contract coherent states, resolutions of the identity, and tight frames. In order to obtain the standard Gabor frame we construct a family of time localized wavelets frames that contract to that Gabor frame. Starting from a standard wavelets frame we construct a family of frequency localized wavelets frames that contract to a nonstandard Gabor frame. In particular we deform Gabor frames to wavelets frames.

math.RT↗

arXiv2017

Newforms of half-integral weight: the minus space counterpart

We define a subspace of the space of holomorphic modular forms of weight $k+1/2$ and level $4M$ where $M$ is odd and square-free. We show that this subspace is isomorphic under the Shimura-Niwa correspondence to the space of newforms of weight $2k$ and level $2M$ and that this is a Hecke isomorphism. The space we define is a proper subspace of the orthogonal complement of the Kohnen plus space if the Kohnen plus space is nonzero.

math.NT↗

arXiv2015

Hecke algebras, new vectors and new forms on $Γ_0(m)$

We characterize the space of new forms for $Γ_0(m)$ as a common eigenspace of certain Hecke operators which depend on primes $p$ dividing the level $m$. To do that we find generators and relations for a $p$-adic Hecke algebra of functions on $K={\rm GL}_2(\mathbb{Z}_p)$. We explicitly find the $n+1$ irreducible representations of $K$ which contain a vector of level $n$ including the unique representation that contains the "new vector" at level $n$. After translating the $p$-adic Hecke operators that we obtain into classical Hecke operators we obtain the results about the new space mentioned above.

math.NT↗

arXiv2012

On the contraction of so(4) to iso(3)

For any skew-Hermitian integrable irreducible infinite dimensional representation $η$ of $iso(3)$, we find a sequence of (finite dimensional) irreducible representations $ρ_n$ of $so(4)$ which contract to $η$.

math-ph↗

arXiv2010

The Classical Hankel Transform In The Kirillov Model

We give a new and simple proof of the Hankel inversion formula for the classical Hankel transform which holds for a complex order with real part greater than -1. Using the proof of this formula we obtain the full description of the Kirillov model for discrete series representations of SL(2,R) and GL(2,R).

math.CA↗

arXiv2003

Central value of automorphic $L-$functions

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

math.NT↗