Invariant Linear Forms Research Articles

The support of closed orbit relative matrix coefficients

Let $F$ be a nonarchimedean local field with odd residual characteristic and let $G$ be the $F$-points of a connected reductive group defined over $F$. Let $\theta$ be an $F$-involution of $G$. Let $H$ be the subgroup of $\theta$-fixed points in $G$. Let $\chi$ be a quasi-character of $H$. A smooth complex representation $(\pi,V)$ of $G$ is $(H,\chi)$-distinguished if there exists a nonzero element $\lambda$ in $\operatorname{Hom}_H(\pi,\chi)$. We generalize a construction of descended invariant linear forms on Jacquet modules first carried out independently by Kato and Takano (2008), and Lagier (2008) to the setting of $(H,\chi)$-distinction. We follow the methods of Kato and Takano, providing a new proof of similar results of Delorme (2010). Moreover, we give an $(H,\chi)$-analogue of Kato and Takano's relative version of the Jacquet Subrepresentation Theorem. In the case that $\chi$ is unramified, $\pi$ is parabolically induced from a $\theta$-stable parabolic subgroup of $G$, and $\lambda$ arises via the closed orbit in $Q\backslash G / H$, we study the (non)vanishing of the descended forms via the support of $\lambda$-relative matrix coefficients.

Linear forms, algebraic cycles, and derivatives of L-series

In this note, we state some refinements of conjectures of Gan-Gross-Prasad and Kudla concerning the central derivatives of L-series and special cycles on Shimura varieties. The analogues of our formulation for special values of L-series are written in terms of invariant linear forms on autormorphic representations defined by integrations of matrix coefficients.

On symplectic periods and restriction to SL(2n)

We study representations of the general and the special linear groups over a non-archimedean local field that admit a non-zero invariant linear form with respect to the symplectic group (henceforth distinguished representations). For the class of ladder representations that are distinguished we show that applying the highest derivative operation twice results in a distinguished (ladder) representation. Furthermore, we characterize the unique distinguished component of the associated L-packet of representations of the special linear group in terms of the associated maximal unipotent orbit. We also obtain a sufficient condition for distinction of standard modules.

Distinction of the Steinberg representation for inner forms of GL(n)

Let F be a non archimedean local field of characteristic not 2. Let D be a division algebra of dimension $$d^2$$ over its center F, and E a quadratic extension of F. If m is a positive integer, to a character $$\chi $$ of $$E^*$$ , one can attach the Steinberg representation $$St(\chi )$$ of $$G=GL(m,D\otimes _F E)$$ . Let H be the group GL(m, D), we prove that $$St(\chi )$$ is H-distinguished if and only if $$\chi _{|F^*}$$ is the quadratic character $$\eta _{E/F}^{md-1}$$ , where $$\eta _{E/F}$$ is the character of $$F^*$$ with kernel the norms of $$E^*$$ . We also get multiplicity one for the space of invariant linear forms.

Test vectors for local periods

Abstract Let E / F {E/F} be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation π of GL ⁢ ( n , E ) {\mathrm{GL}(n,E)} is said to be distinguished with respect to GL ⁢ ( n , F ) {\mathrm{GL}(n,F)} if it admits a non-trivial linear form that is invariant under the action of GL ⁢ ( n , F ) {\mathrm{GL}(n,F)} . It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [14] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

Modular Representations of ${\rm GL}{(n)}$ Distinguished by ${\rm GL}{(n-1)}$ over a $p$-Adic Field

Let F be a non-Archimedean locally compact field, q be the cardinality of its residue field, and R be an algebraically closed field of characteristic l not dividing q⁠. We classify all irreducible smooth R-representations of GLn(F) having a non-zero GLn−1(F)-invariant linear form, when q is not congruent to 1 mod l⁠. Partial results in the case when q is 1 mod l show that, unlike the complex case, the space of GLn−1(F)-invariant linear forms has dimension 2 for certain irreducible representations..

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.

Traces on operator ideals and related linear forms on sequence ideals (part III)

Let S+ be the forward shift acting on sequences indexed by N0:={0,1,2,…}. With certain sequence ideals z(N0) we associate an operator ideal Dzapp over the class of all Banach spaces. It will be proved that every 12S+-invariant linear form λ on z(N0) generates a trace on Dzapp, which is presented by an explicit expression. If the considerations are restricted to operators on the infinite-dimensional separable Hilbert space, then our elementary approach yields all traces on all operator ideals.

Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part II)

The theory of operator ideals on the infinite-dimensional separable Hilbert space is equivalent to the theory of shift-monotone sequence ideals. In particular, traces correspond to $${\frac {1}{2} S_+}$$ -invariant linear forms, where S + is the forward shift. Important consequences of these facts are presented. We concentrate on the study of sequence ideals, which are easier to handle. So the reader should not be surprised that operator ideals play almost no role. All results, however, can be translated into the context of operators. In this way, algebraic and topological classifications of operator ideals are obtained.

Galois ring extensions and localized modular rings of invariants of p-groups

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].

Traces on operator ideals and related linear forms on sequence ideals (part I)

The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}.The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1)For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh.(2)z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25]The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0).In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”.Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.

On representations distinguished by unitary groups

Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on \(\operatorname {GL}_{n}( \mathbf {A}_{E})\) over a unitary group of a Hermitian form with respect to E/F. We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and characterize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group.

Distinguished Generic Representations of GL(n) over p-adic Fields

Let K/F be a quadratic extension of p-adic fields, and n a positive integer. A smooth irreducible representation of the group GL(n, K) is said to be distinguished, if it admits on its space a nonzero GL(n, F)-invariant linear form. In the present work, we classify distinguished generic representations of the group GL(n, K) in terms of inducing quasi-square-integrable representations. This has, as a consequence, the truth of the expected equality between the Rankin–Selberg type Asai L-function of a generic representation and the Asai L-function of its Langlands parameter.

Generalised form of a conjecture of Jacquet and a local consequence

Following the work of Harris and Kudla, we prove a general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain L-function. As a consequence, we deduce a theorem relating the existence of GL 2 (K) -invariant linear forms on irreducible, admissible representations of GL 2 (K) for a commutative semi-simple cubic algebra K over a non-archimedean local field k in terms of local epsilon factors which was proved only in some cases by the first author in his earlier work in [D. Prasad, Invariant forms for representations of GL 2 over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363.]. This has been achieved by globalising a locally distinguished supercuspidal representation to a globally distinguished representation, a result of independent interest which is proved by an application of the relative trace formula.

Distinguished representations for $SL(2)$

Let E/F be a quadratic extension of p-adic fields. We compute the multiplicity of the space of SL 2 (F)-invariant linear forms on a representation of SL 2 (E). This multiplicity varies inside an L-packet similar in spirit to the multiplicity formula for automorphic representations due to Labesse and Langlands.

Cuspidal Representations Associated to (GL(n), O(n)) over Finite Fields andp-Adic Fields

Whenr̄is an irreducible cuspical representation ofḠ=GL(n,q) andH̄is an orthogonal group associated to a symmetric matrix inḠthen the space ofH̄-fixed vectors forr̄is shown to have dimension at most one. Such a representationr̄induces an irreducible supercuspidal representation π ofG=GL(n,E), whereEis ap-adic field whose residue field has orderq. The space of those linear forms on the space of π which are invariant under an orthogonal group is computed. For the corresponding group of orthogonal similitudes, it is shown that the dimension of the space of invariant linear forms is always at most one.

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.

Some Remarks on Representations of a Division Algebra and of the Galois Group of a Local Field

In this paper we prove a division algebra analogue of a theorem of Jacquet and Rallis about uniqueness ofGLn(k)×GLn(k) invariant linear form on an irreducible admissible representation ofGL2n(k). We propose a conjecture about when this invariant form exists. We prove some results about self-dual representations of the invertible elements of a division algebra and of Galois groups of local fields. The Shalika model has been studied for principal series representations ofGL2(D) forDa division algebra and a conjecture made regarding its existence in general.

Supercuspidal representations of GL(n) distinguished by a unitary subgroup

When E=F is a quadratic extension of p-adic elds, with p6 , and H 0 is a unitary similitude group in GL(n;E) ,i t is shown that for every irreducible supercuspidal representation of GL(n;E) of lowest level the space of H 0 -invariant linear forms has dimension at most one. The analogous fact for the corresponding unitary group H also holds, so long as n is odd or E=F is ramied. When n is even and E=F is unramied, the space of H-invariant linear forms on the space of may have dimension two.

On Error-Correcting Codes and Invariant Linear Forms

Given a code C, invariant linear forms are used to study the designs afforded by codewords of a fixed weight. The most important theorem relating codes and designs is due to Assmus and Mattson [J. Combin. Theory, 6 (1969), pp. 122–151], and this theorem is extended in different ways. For extremal self dual codes over the fields $\mathbb{F}_2 $ and $\mathbb{F}_3 $, it is proved that the t-designs afforded by the codewords of any fixed weight exhibit extra regularity with respect to $( t + 2 )$-sets. The same is true for the design afforded by the codewords of minimum weight in an extremal self-dual code over $\mathbb{F}_4 $. The invariant linear forms are also used to construct Boolean designs with several block sizes, extending previous work by Safavi-Naini and Blake [Utilitas Math., 14 (1978), pp. 49–63], [Ars Combin., 7 (1979), pp. 135–151], [Inform. and Control, 42 (1986), pp. 261–282].