Doubling constructions and tensor product L-functions: coverings of the symplectic group

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadic. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type AB and D and establish functoriality properties, relating categories with disjoint I’s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato’s exotic geometry.

Compactness in dual spaces of locally compact groups and tensor products of irreducible representations

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition.

The tensor product algebra [Formula: see text] for the complex general linear group [Formula: see text], introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of [Formula: see text]. Using the hive model for the Littlewood–Richardson (LR) coefficients, we provide a finite presentation of the algebra [Formula: see text] for [Formula: see text] in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of LR coefficients.

If G is a reductive group in characteristic zero with two irreducible representations V1 and V2, we may form the tensor product representation V1 0 V2 where G acts diagonally. The problem is to decompose V1 0 V2 into its irreducible components with their multiplicities. Of course this may be done using Weyl's character formula but this is a very difficult calculation. This paper presents a general method for doing this calculation in a more efficient way. For the general linear group the method was explained in [2] with a minimum of group-theoretic language. For the other classical groups this paper will give explicit formulas. The method of this paper may be used for exceptional groups but for combinatorial reasons it does not seem worthwhile at this point to write an explicit formula. Perhaps it would be more useful to have a computer program for the calculation. As an after thought I have included an appendix on how to decompose an irreducible representation of G when restricted to a large reductive subgroup. The case of the general linear group is well-known.

For each of the exceptional Lie groups, a complete determination is given of those pairs of finite-dimensional irreducible representations whose tensor products (or squares) may be resolved into irreducible representations that are multiplicity free, i.e. such that no irreducible representation occurs in the decomposition of the tensor product more than once. Explicit formulae are presented for the decomposition of all those tensor products that are multiplicity free, many of which exhibit a stability property.

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.

We first determine the kernels of those unitary representations of a locally compact group which are obtained by integrating, inducing and tensoring. We then describe a broad class of group extensions for which the previous results can be used to find (1) the kernel of any representation and (2) necessary and sufficient conditions for these extensions to be maximally almost periodic. Introduction. The primary purpose of this paper is to determine the kernel of any (unitary) representation of a reasonably general locally compact group extension. (See ?4 for a specific description of the exten- sion.) Roughly speaking, the representations of such an extension are obtained by successively tensoring, inducing and integrating (at least in the separable case) representations of certain subgroups. In ?1 we deter- mine the kernel of a direct integral. (This is the only place where we find it necessary to introduce separability assumptions.) This determination has the effect of reducing our problem to the consideration of irreducible representations only. In ?2 we describe the kernel of an induced repre- sentation, thus extending Lemma 2.1 of (11) to the nonseparable case. Regarding the tensor product of representations, it is well known (see (9) and (6, ?17)) that the most useful kernel to determine is that of the tensor product of projective representations or, more accurately, of representa- tions of corresponding central group extensions of the circle. We do this in ?3. Finally, in ?4 we describe the type of extension for which the preceding results solve the problem stated at the beginning. This section is essentially a combination and augmentation of R. J. Blattner's work in (3) and the relevant results of J. M. G. Fell in ?17 of (6). We conclude by indicating how the previous results can be used to find necessary and sufficient conditions for the extension to be maximally almost periodic. Throughout this paper, G will be a locally compact group and all representations will be unitary. If h is a group homomorphism then

We study tensor product decompositions of representations of the General Linear Supergroup Gl(m|n). We show that the quotient of Rep(Gl(m|n),\epsilon)$ by the tensor ideal of negligible representations is the representation category of a pro-reductive supergroup G red. In the Gl(m|1)-case we show G red = Gl(m-1) \times Gl(1) \times Gl(1). In the general case we study the image of the canonical tensor functor Fmn from Deligne's interpolating category Rep (Gl m-n) to Rep(Gl(m|n),\epsilon). We determine the image of indecomposable elements under Fmn. This implies tensor product decompositions between projective modules and between certain irreducible modules, including all irreducible representations in the Gl(m|1)-case. Using techniques from Deligne's category we derive a closed formula for the tensor product of two maximally atypical irreducible Gl(2|2)-representations. We study cohomological tensor functors DS : Rep(Gl(m|m), epsilon) -> Rep(Gl(m-1|m-1)) and describe the image of an irreducible element under DS. At the end we explain how these results can be used to determine the pro-reductive group G L \hookrightarrow Gl(m|m) red corresponding to the subcategory Rep(G L, epsilon) generated by the image of an irreducible element L in Rep(Gl(m|m) red, epsilon).

Following a recent proposal of Richard Borcherds to regard fusion as the ringlike tensor product of modules of a quantum ring, a generalization of rings and vertex algebras, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra [Formula: see text]. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of [Formula: see text] on it, under which the central extension is preserved. Having defined fusion in this way, determining the fusion rules is then the algebraic problem of decomposing the tensor product into irreducible representations. We demonstrate how to solve this for the case of the WZW and the minimal models and recover thereby the well-known fusion rules. The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the R matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible rôle of the quantum group in conformal field theory.

Tensor products of maximal degenerate series representations of the group SL( n, R)

Decomposing tensor products of irreducible representations of compact groups almost always involves multiplicity, wherein some irreducible representations occur more than once in the direct sum decomposition. We show that the multiplicity can always be specified by polynomial group invariants. The setting is a Bargmann–Segal–Fock space in n×N complex variables, where n is the number of labels needed to specify the tensor product and N is the dimension of the fundamental representation of the compact group. Both the tensor product and direct sum bases are realized as polynomials in this space, and it is shown how Clebsch–Gordan and Racah coefficients can be computed by suitably differentiating these polynomials. The example of SU(N) is discussed in detail, and it is shown that the multiplicity can be computed as the solution of certain diophantine equations arising from powers of group invariants, namely minors of determinants.

Let $GL_M$ be general linear Lie group over the complex field. The irreducible rational representations of the group $GL_M$ are labeled by pairs of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero parts of $\mu$ and $\tilde{\mu}$ does not exceed $M$. Let $U$ be the representation of $GL_M$ corresponding to such a pair. Regard the direct product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the irreducible rational representation of the group $GL_{N+M}$ corresponding to a pair of partitions $\lambda$ and $\tilde{\lambda}$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $GL_N$. Let $n$ be sum of parts of $\lambda$ less the sum of parts of $\mu$. Let $\tilde{n}$ be sum of parts of $\tilde{\lambda}$ less the sum of parts of $\tilde{\mu}$. For any choice of two standard Young tableaux of skew shapes $\lambda/\mu$ and $\tilde{\lambda}/\tilde{\mu}$ respectively, we realize $W$ as a subspace in the tensor product of $n$ copies of the defining $N$-dimensional representation of $GL_N$, and of $\tilde{n}$ copies of the contragredient representation. This subspace is determined as the image of a certain linear operator $F$ in the tensor product, given by explicit multiplicative formula. When M=0 and $W=V$ is an irreducible representation of $GL_N$, we recover the classical realization of $V$ as a subspace in the space of all traceless tensors. Then the operator $F$ can be regarded as the rational analogue of the Young symmetrizer, corresponding to the chosen standard tableau of shape $\lambda$. Even in the special case M=0, our formula for the operator $F$ is new. Our results are applications of representation theory of the Yangian of the Lie algebra $gl_N$.

There is a natural bijective correspondence between irreducible (algebraic) selfdual representations of the special linear group with those of classical groups. In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of $$\mathrm{SL}_{2n}({\mathbb {C}})$$ with those of $$\mathrm{Spin}_{2n+1}({\mathbb {C}})$$ , it is easy to see that if the tensor product of three irreducible representations of $$\mathrm{Spin}_{2n+1}({\mathbb {C}})$$ contains the trivial representation, then so does the tensor product of the corresponding representations of $$\mathrm{SL}_{2n}({\mathbb {C}})$$ . The paper formulates a conjecture in the reverse direction for the pairs $$(\mathrm{SL}_{2n}({\mathbb {C}}), \mathrm{Spin}_{2n+1}({\mathbb {C}})), (\mathrm{SL}_{2n+1}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})),$$ and $$ (\mathrm{Spin}_{2n+2}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})) $$ .

To each Levi subgroup of a general linear group there corresponds a set of general linear groups of smaller order. One may therefore construct an irreducible representation of such a Levi subgroup by taking the tensor product of irreducible representations of the smaller general linear groups. We generalize this construction to the context of metaplectic coverings over a p-adic field.