Arbitrary Semisimple Group Research Articles

Essential dimension of semisimple groups of type B

We determine the essential dimension of an arbitrary semisimple group of type B of the formG=(Spin(2n1+1)×⋯×Spin(2nm+1))/μ over a field of characteristic 0, for all n1,…,nm≥7, and a central subgroup μ of Spin(2n1+1)×⋯×Spin(2nm+1) not containing the center of Spin(2ni+1) as a direct factor for every i=1,…,m.

The moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory

We determine the dimension of the moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory in 2 + 1 dimensions for gauge groups with G′ being an arbitrary semi-simple group and n 0 the greatest common divisor of the abelian charges of the G′ invariants. The calculation is carried out using a Callias-type index theorem, the moduli matrix approach and a D-brane setup in type IIB string theory. We prove that the index theorem gives the number of zeromodes or moduli of the non-abelian vortices, extend the moduli matrix approach to the Yang–Mills–Chern–Simons–Higgs theory and finally derive the effective Lagrangian of Collie and Tong using string theory.

EXTREMAL RAYS IN THE HERMITIAN EIGENVALUE PROBLEM FOR ARBITRARY TYPES

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G = SL(n). There is a polyhedral cone (the \eigencone) determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G. We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face. The remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [B3] which handled the case of G = SL(n).

A notion of rank for unitary representations of reductive groups based on Kirillov's orbit method

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe's theory of N-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors' works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov's theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S]

Zariski-dense subgroups and transcendental number theory

Let K be a field, L be a field extension of K, and G be a connected semisimple algebraic group defined over K. In [6], we called a K-torus T to be K-irreducible if it does not contain any proper K-subtori. We established the existence of such tori in any connected absolutely (almost) simple algebraic group over a global field and gave applications of this result, in particular, to the congruence subgroup problem. Subsequently, in [8] we considered maximal tori in arbitrary semisimple groups, where the notion of irreducible tori had to be replaced with a more general notion of quasi-irreducible tori. As in [8], we will say that a maximal L-torus T of G is L-quasi-irreducible (or, quasi-irreducible over L) if it does not contain any L-subtori other than (almost direct) products of the subtori T (i) := T ∩G(i), where G, . . . , G are the connected L-simple normal subgroups of G. In [8], we proved the existence of K-quasi-irreducible tori in G if K is a finitely generated field of characteristic zero. (A noteworthy feature of the argument in [8] is that it is based on the technique of embedding relevant finitely generated fields into nonarchimedean local fields, developed and used earlier in [7] for a different purpose.) Assume now that K is a finitely generated subfield of the field R of real numbers. In [8] we proved that any Zariski-dense subsemigroup of G(K) contains a regular R-regular element x such that the centralizer T of x in G is a K-quasiirreducible torus and the cyclic subgroup of T (K) generated by x is Zariski-dense in T . Since such elements will also play an important role in the current paper, we recall for the reader’s convenience that a semisimple element x ∈ G is called regular if the identity component ZG(x)◦ of its centralizer is a (maximal) torus; note that x ∈ ZG(x)◦ ([4], 11.12). Regular elements form a Zariski-open subset of G (cf. [4], §12). A regular element x of G(L) will be called L-quasi-irreducible if the maximal L-torus T := ZG(x)◦ is L-quasi-irreducible, and x will be called L-quasi-irreducible anisotropic if, in addition, T is anisotropic over L. We will say that an element x ∈ G(K) is without components of finite order, if in some (equivalently, any) decomposition x = x1 · · ·xt, with xi ∈ Gi, where G1, . . . ,Gt are the connected absolutely simple normal subgroups of G, all the xi’s have infinite order. It is easy to see that (for x ∈ G(K)) this notion is equivalent to the one introduced in [8]. An element x ∈ G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. Such an element

Total positivity in Schubert varieties

We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups.

On a problem of Pommerening in invariant theory

Hochschild and Mostow [3] have proved the finite generation of the algebra of invariants for the regular action of the unipotent radical of a parabolic subgroup on the polynomial functions on a complex semisimple algebraic group G. Pommerening [S] used combinatorial techniques to show that if G = SL, then we may remove any set of simple roots from the unipotent radical, and the algebra of invariants will still be finitely generated. In this paper we partially generalize Pommerening’s theorem to arbitrary semisimple groups G. In the proof we classify in terms of Dynkin diagrams, the cases for which the results of Grosshans [2] are applicable. Let R be the root system relative to a maximal torus T of G, and let D be a basis of R (which we identify with the Dynkin diagram of R). For a closed subset S of the set R + of positive roots, we denote by Us the corresponding standard unipotent subgroup of G, that is, U, = naES U,, where the product is taken in any fixed order and the U, are the root groups. For a subset I of D we denote S1 = R ‘\Z’I (so that U,, is the unipotent radical of a parabilic subgroup). Denote by C[G] the algebra of polynomial functions on G. The main result of this paper is the following.

Reductive Groups are Geometrically Reductive

Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of invariants, RG, is finitely generated originates with the invariant theorists of the nineteenth century. When k = C, the complex numbers, and G GL (n, C) the question is answered affirmatively by Hilbert's fundamental theorem of invariant theory. The proof involved constructing a G equivariant projection from R to RG and then using it to prove the result algebraically. When k is of characteristic 0 and G is any semi-simple group, by a theorem of H. Weyl, every finite dimensional representation of G is completely reducible. In the 1950's D. Mumford and others (Cartier, Iwahori, Nagata) applied Weyl's theorem to construct a projection from R to RG for any semi-simple group. This made it possible to generalize Hilbert's proof to an arbitrary semi-simple group. Certain geometric applications, particularly to the theory of moduli, made a generalization to groups over fields of positive characteristic highly desirable. In positive characteristic, complete reducibility definitely fails. Hence attempts were made to replace complete reducibility with a weaker condition which would at once hold for all semi-simple groups and make a proof of finite generation of RG possible. The weakest way to state complete reducibility is the following. If V is a finite dimensional G-module containing a G-stable sub-space of co-dimension one, VT, then there is a G-stable line LC V such that V0 E L = V. Mumford conjectured a weaker version of this statement by seeking a complement only in a higher symmetric power of V, SI( V). This is the conjecture as it is stated in the preface to [16]:

Equivalent conditions for the 𝐿_{𝑝} convolution theorem on semisimple groups

For certain semisimple Lie groups G, it is known that convolution by an L p ( G ) {L_p}(G) function, 1 ≦ p > 2 1 \leqq p > 2 , is a bounded operator on L 2 ( G ) {L_2}(G) . This result is a consequence of the so-called “analytic continuation of the principal series” which has been carried out on these groups. However, this continuation procedure does not generalize readily to arbitrary semisimple groups. In an attempt to bypass the continuation and obtain the convolution theorem in an alternate manner, we derive in this paper several equivalent conditions for this L p {L_p} convolution theorem.