Irreducible Summands Research Articles

On automorphic descent from $GL_7$ to $G_2$

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations $\pi$ of $GL\_7(\mathbb{A})$ with $L^S(s, \pi, \wedge^3)$ having a pole at $s=1$ to the split exceptional group $G\_2(\mathbb{A})$, using Fourier coefficients associated to two nilpotent orbits of $E\_7$. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and $\pi$ is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube $L$-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on $G\_2(\mathbb{A})$ but also every degenerate Whittaker–Fourier integral. Thus it is generic, but not cuspidal.

Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties

We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kähler package.”

Left-Invariant Einstein-like Metrics on Compact Lie Groups

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We prove that the left metric ⟨·,·⟩t1,t2 on G defined by the first equation, must be an A-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B-metrics, such as ⟨·,·⟩t1,t2.

Rank 2 local systems and abelian varieties II

Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal {E}_i$ of $\mathcal {E}$ have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal {E}$ at $x$ is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$. Then there exists a dense open subset $U\subset X$ such that $\mathcal {E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar {\mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar {\mathbb {Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$.

Boundary values in spaces spanned by rational functions and the index of invariant subspaces

Let μ $\mu$ be a positive compactly supported measure in the complex plane C $\mathbb {C}$ , and for each p , 1 ⩽ p < ∞ $p,1\leqslant p<\infty$ , let H p ( μ ) $H^p(\mu )$ be the closed subspace of L p ( μ ) $L^p(\mu )$ spanned by the polynomials. In 1991, Thomson gave a complete description of its structure, expressing H p ( μ ) $H^p(\mu )$ as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non-trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to R p ( μ ) $R^p(\mu )$ , the closed subspace of L p ( μ ) $L^p(\mu )$ spanned by the rational functions having no poles on the support of μ $\mu$ , by first establishing the existence of boundary values in these spaces. Our results all depend on the semiadditivity of analytic capacity, and ultimately on some form of the F. and M. Riesz theorem.

Multiplicity in root components via geometric Satake

In this note we explicitly construct top-dimensional components of certain cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands for where and β is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when β is not a simple root. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.

On the multiplication of spherical functions of reductive spherical pairs of type A

AbstractLetGbe a simple complex algebraic group, and let$K \subset G$be a reductive subgroup such that the coordinate ring of$G/K$is a multiplicity-freeG-module. We consider theG-algebra structure of$\mathbb C[G/K]$and study the decomposition into irreducible summands of the product of irreducibleG-submodules in$\mathbb C[G/K]$. When the spherical roots of$G/K$generate a root system of type$\mathsf A$, we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of$G/K$is a direct sum of subsystems of rank 1.

Sigma involutions associated with parafermion vertex operator algebra

An irreducible module for the parafermion vertex operator algebra is said to be of σ-type if an automorphism of the fusion algebra of of order k is trivial on it. For any integer k ≥ 3, we show that there exists an automorphism of order 2 of the subalgebra of the fusion algebra of spanned by the irreducible direct summands of σ-type irreducible -modules, where θ is an involution of . We discuss some examples of such an automorphism as well.

Branching Problems of Unitary Representations

The irreducible decomposition of a unitary representation often contains con­ tinuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs discretely with finite multiplicity (admissible restrictions). Basic theory and new perspectives of admissible restrictions are presented from both analytic and algebraic view points. We also discuss some applications of admissible restrictions to modular varieties and Lp-harmonic analysis.

Two boundary centralizer algebras for [formula omitted

We define the degenerate two boundary affine Hecke-Clifford algebra Hd, and show it admits a well-defined q(n)-linear action on the tensor space M⊗N⊗V⊗d, where V is the natural module for q(n), and M,N are arbitrary modules for q(n), the Lie superalgebra of Type Q. When M and N are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of Hd. We then construct explicit modules for this quotient, Hp,d, using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in Hp,d, we also classify a specific class of calibrated modules. The irreducible summands of M⊗N⊗V⊗d coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.

The self-coupled Einstein–Cartan–Dirac equations in terms of Dirac bilinears

In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann–Cartan spacetime with torsion; the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein–Cartan–Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit rational expression of the four-vector potential of the gauge field in terms of the spinors. Substitution of this expression into Maxwell’s equations yields the bilinear form of the self-interactive Maxwell–Dirac equations. In the present (purely gravitational) case, the inversion process yields two rational four-vector expressions in terms of Dirac bilinears, which act as gravitational analogues of the electromagnetic vector potential. These ‘potentials’ also appear as irreducible summand components of the connection, along with a traceless residual term of mixed symmetry. When taking the torsion field equation into account, the residual term can be written as a function of the object of anholonomity. Using the local tetrad frame associated with observers co-moving with the Dirac matter, a generic vierbein frame can described in terms of four Dirac bilinear vector fields, normalized by a scalar and pseudoscalar field. A corollary of this is that in regions where the Dirac field is non-vanishing, the self-coupled Einstein–Cartan–Dirac equations can in principle be expressed in terms of Dirac bilinears only.

LINEAR RECURRENCE RELATIONS IN Q-SYSTEMS VIA LATTICE POINTS IN POLYHEDRA

We prove that the sequence of the characters of the Kirillov–Reshetikhin (KR) modules $$ {W}_m^{(a)} $$ , m ∈ ℤm≥0 associated to a node a of the Dynkin diagram of a complex simple Lie algebra $$ \mathfrak{g} $$ satisfies a linear recurrence relation except for some cases in types E7 and E8. To this end we use the Q-system and the existing lattice point summation formula for the decomposition of KR modules, known as domino removal rules when $$ \mathfrak{g} $$ is of classical type. As an application, we show how to reduce some unproven lattice point summation formulas in exceptional types to finite problems in linear algebra and also give a new proof of them in type G2, which is the only completely proven case when KR modules have an irreducible summand with multiplicity greater than 1. We also apply the recurrence to prove that the function dim $$ {W}_m^{(a)} $$ is a quasipolynomial in m and establish its properties. We conjecture that there exists a rational polytope such that its Ehrhart quasipolynomial in m is dim $$ {W}_m^{(a)} $$ and the lattice points of its m-th dilate carry the same crystal structure as the crystal associated with $$ {W}_m^{(a)} $$ .

Existence of homogeneous metrics with prescribed Ricci curvature

Consider a compact Lie group G and a closed subgroup H<G. Suppose T is a positive-definite G-invariant (0,2)-tensor field on the homogeneous space M=G/H. In this note, we state a sufficient condition for the existence of a G-invariant Riemannian metric on M whose Ricci curvature coincides with cT for some c>0. This condition is, in fact, necessary if the isotropy representation of M splits into two inequivalent irreducible summands. After stating the main result, we work out an example.

Structure of the Main Tensor of Conformally Connected Torsion Free Space. Conformal Connections on Hypersurfaces of Projective Space

We give a definition of a conformally connected space with an angular metric of an arbitrary signature. We present basic formulas and classes of such spaces. We obtain the decomposition of the main tensor of a conformally connected torsion free space into irreducible gauge invariant summands and prove that all affine connections obtained from the Levi–Civita connection via the normalization transformation have the same Weyl conformal tensor. We describe all conformal torsion free connections on hypersurfaces of a projective space and give some examples. We construct a global conformal connection on a hyperquadric of the projective space.

The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Let be a compact homogeneous space, and let and be G-invariant Riemannian metrics on . We consider the problem of finding a G-invariant Einstein metric g on the manifold subject to the constraint that g restricted to and coincides with and , respectively. By assuming that the isotropy representation of consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.

Realization of U&lt;sub&gt;q&lt;/sub&gt;(sp&lt;sub&gt;2n&lt;/sub&gt;) within the Differential Algebra on Quantum Symplectic Space

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a $U_q({\mathfrak{sp}}_{2n})$-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of $U_q({\mathfrak {sp}}_{2n})$.

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.

Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

We find the algebras of unipotent invariants of the Cox rings for all double flag varieties of complexity and for the exceptional simple algebraic groups; namely, we obtain presentations of these algebras in terms of generators and relations. It is well known that such an algebra is free in the case of complexity . In this paper, we show that, in the case of complexity , the algebra in question is either a free algebra or a hypersurface. A similar result for classical groups was previously obtained by the author. Knowing the structure of this algebra enables one to decompose tensor products of some irreducible representations effectively into irreducible summands and to obtain some branching rules. Bibliography: 10 titles.

Twisted symmetric square L-functions for $$\mathrm {GL}_n$$ GL n and invariant trilinear forms

Following the works of Bump and Ginzburg and of Takeda, we develop a theory of twisted symmetric square L-functions for \(\mathrm {GL}_n\). We characterize their pole in terms of certain trilinear period integrals, determine all irreducible summands of the discrete spectrum of \(\mathrm {GL}_n\) having nonvanishing trilinear periods, and construct nonzero local invariant trilinear forms on a certain family of induced representations.

Polynomial Solutions For Arbitrary Higher Spin Dirac Operators

In a series of recent papers, we have introduced higher spin Dirac operators, which are far-reaching generalisations of the classical Dirac operator. Whereas the latter acts on spinor-valued functions, the former acts on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe a general procedure to decompose the polynomial kernel spaces for these operators in irreducible summands for the regular action of the spin group. We will do this in an inductive way, making use of twisted higher spin operators.