Complex Geometry and Representations of Lie Groups
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Double Coset Formulas for Profinite Groups 1
DOUBLE COSET FORMULAS FOR PROFINITE GROUPS PETER SYMONDS Abstract. We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups. 1. Introduction The double coset formula, which expresses the restriction of an induced module as a direct sum of induced modules, is a basic tool in the representation theory of groups, but it is not always valid as it stands for profinite groups. Based on our work on permutation modules in [8], we give some sufficient conditions for a strong form of the formula to hold. These conditions are not necessary, although they do often hold in interesting cases, and at the end we give some simple examples where the formula fails. We also formulate a weaker version of the formula that does hold in general. 2. Results We work with profinite groups G and their representations over a complete commutative noetherian local ring R with finite residue class field of characteristic p. There is no real loss ˆ of generality in taking R = Zp. Our representations will be in one of two categories: the discrete p-torsion modules DR(G) or the compact pro-p modules CR(G). These are dual by the Pontryagin duality functor Hom(−, Q/Z), which we denote by ∗, so we will usually work in DR(G). Our modules are normally left modules, so dual means contragredient. For more details see [7, 8]. For any profinite G-set X and M ∈ DR(G) we let F (X, M) ∈ DR(G) denote the module of continuous functions X → M, with G acting according to (gf)(x) = g(f(g−1x)), g ∈ G, f ∈ F (X, M), x ∈ X. -
Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type
HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE JURGEN¨ BERNDT Abstract. A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jos´eCarlos D´ıaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1. Isometric actions on the real hyperbolic plane The special linear group a b G = SL2(R) = a, b, c, d ∈ R, ad − bc = 1 c d acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form a b az + b · z = . c d cz + d By equipping H with the Riemannian metric dx2 + dy2 dzdz¯ ds2 = = (z = x + iy) y2 =(z)2 we obtain the Poincar´ehalf-plane model of the real hyperbolic plane RH2. The above action of SL2(R) is isometric with respect to this metric. The action is transitive and the stabilizer at i is cos(s) sin(s) K = s ∈ R = SO2. -
Invariant Differential Operators for Quantum Symmetric Spaces, I
Invariant Differential Operators for Quantum Symmetric Spaces, I Gail Letzter∗ Mathematics Department Virginia Tech Blacksburg, VA 24061 [email protected] January 1, 2018 Abstract This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential op- erators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we establish this result for all quantum symmetric spaces defined using irreducible sym- metric pairs not of type EIII, EIV, EVII, or EIX. A quantum version of a related theorem due to Helgason is also obtained: The image of the center under this Harish-Chandra map is the entire invariant ring arXiv:math/0406193v1 [math.QA] 9 Jun 2004 if and only if the underlying irreducible symmetric pair is not one of these four types. Introduction In [HC2, Section 4], Harish-Chandra proved the following fundamental result for semisimple Lie groups: the Harish-Chandra map induces an isomorphism ∗supported by NSA grant no. MDA904-03-1-0033. AMS subject classification 17B37 1 between the ring of invariant differential operators on a symmetric space and invariants of an appropriate polynomial ring under the restricted Weyl group. When the symmetric space is simply a complex semisimple Lie group, this result is Harish-Chandra’s famous realization of the center of the enveloping algebra of a semisimple Lie algebra as Weyl group invariants of the Cartan subalgebra ([HC1]). -
Decomposition Theorems for Automorphism Groups of Trees
DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES MAX CARTER AND GEORGE WILLIS Abstract. Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, double cosets of the group of label preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups are enumerated. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range. 1. Introduction Totally disconnected locally compact groups (t.d.l.c. groups here on after) are a broad class which includes Lie groups over local fields. It is not known, however, just how broad this class is, and one approach to investigating this question is to attempt to extend ideas about Lie groups to more general t.d.l.c. groups. This approach has (at least partially) motivated works such as [4, 8, 9]. That is the approach taken here for Bruhat and Cartan decompositions of gen- eral linear groups over local fields, which are respectively double-coset enumerations of GLpn, Qpq over the subgroups, B, of upper triangular matrices, and GLpn, Zpq, of matrices with p-adic integer entries. Such decompositions of real Lie groups are given in [11, §§VII.3 and 4], and of Lie groups over local fields in [2, 3, 7, 10]. Automorphism groups of regular trees have features in common with the groups GLp2, Qpq (in particular, PGLp2, Qpq has a faithful and transitive action on a reg- ular tree, [12, Chapter II.1]). Here, we find double coset enumerations of automor- phism groups of trees that correspond to Bruhat and Cartan decompositions. -
Separability Properties of Free Groups and Surface Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 78 (1992) 77-84 77 North-Holland Separability properties of free groups and surface groups G.A. Niblo School of Mathemuticul Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS, United Kingdom Communicated by K.W. Gruenberg Received II March 1991 Abstruct Niblo, G.A., Separability properties of free groups and surface groups, Journal of Pure and Applied Algebra 78 (1992) 77-84 A subset X of a group G is said to be separable if it is closed in the profinite topology. Separable subgroups are very useful in low-dimensional topology and there has been some interest in separable double cosets. A new method for showing that a double coset is separable is introduced and it is used to obtain a short proof of the result of Gitik and Rips, that in a free group every double cosct of finitely generated subgroups is separable. In addition it is shown that this property is shared by Fuchsian groups and the fundamental groups of Seifert fibred 3-manifolds. 1. Introduction Let G be a group regarded as acting on itself by left and right multiplication. In order to define a topology on G which is equivariant with respect to both of these actions, we only need to specify a base 93 for the neighbourhoods of the identity element. Definition. Let !93 be the base consisting of the finite-index normal subgroups of G. -
LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
Modular Forms and Hecke Operators
MODULAR FORMS AND HECKE OPERATORS SAMANDA (YUTING) HU Abstract. This paper focuses on discussing Hecke operators in the theory of modular forms and its relation to Hecke rings which occur in representation theory. We will first introduce the basic definitions and properties of modular forms and Hecke operators. In particular, we will show that the space of cusp forms of an arbitrary weight with respect to a specific congruence subgroup is an inner product space and discuss that the space has an orthogonal basis of simultaneous eigenvectors for the Hecke operators. We will then shift the focus to Hecke rings and do relevant computations regarding the Hecke ring of GL2(Qp). Contents 1. Introduction 1 2. Modular Form and Congruence Subgroup 2 + 3. Action of GL2 (Q) on H 3 4. Hecke Operators 4 4.1. Double Coset Operator 4 4.2. hdi and Tp operator 6 4.3. Petersson Inner Product and Adjoints of Hecke Operators 11 5. Another Interpretation of Hecke Algebra 13 5.1. Hecke Ring 14 5.2. Satake Isomorphism 20 Acknowledgement 23 References 23 1. Introduction Modular forms are functions on the complex upper half plane that satisfy cer- tain transformation conditions and holomorphy conditions. The theory of modular forms is a significant branch of modern number theory { for instance, the Modu- larity Theorem suggests that all rational elliptic curves arise from modular forms with some restrictions. This theorem was essential to the proof of Fermat's Last Theorem by Richard Taylor and Andrew Wiles. In this paper, we aim to discuss Hecke operators in the theory of modular forms. -
Lie Groups and Linear Algebraic Groups I. Complex and Real Groups Armand Borel
Lie Groups and Linear Algebraic Groups I. Complex and Real Groups Armand Borel 1. Root systems x 1.1. Let V be a finite dimensional vector space over Q. A finite subset of V is a root system if it satisfies: RS 1. Φ is finite, consists of non-zero elements and spans V . RS 2. Given a Φ, there exists an automorphism ra of V preserving Φ 2 ra such that ra(a) = a and its fixed point set V has codimension 1. [Such a − transformation is unique, of order 2.] The Weyl group W (Φ) or W of Φ is the subgroup of GL(V ) generated by the ra (a Φ). It is finite. Fix a positive definite scalar product ( , ) on V invariant 2 under W . Then ra is the reflection to the hyperplane a. ? 1 RS 3. Given u; v V , let nu;v = 2(u; v) (v; v)− . We have na;b Z for all 2 · 2 a; b Φ. 2 1.2. Some properties. (a) If a and c a (c > 0) belong to Φ, then c = 1; 2. · The system Φ is reduced if only c = 1 occurs. (b) The reflection to the hyperplane a = 0 (for any a = 0) is given by 6 (1) ra(v) = v nv;aa − therefore if a; b Φ are linearly independent, and (a; b) > 0 (resp. (a; b) < 0), 2 then a b (resp. a + b) is a root. On the other hand, if (a; b) = 0, then either − a + b and a b are roots, or none of them is (in which case a and b are said to be − strongly orthogonal). -
The Absolute Continuity of Convolution Products of Orbital Measures In
The Absolute continuity of convolution products of orbital measures in exceptional symmetric spaces K. E. Hare and Jimmy He Abstract. Let G be a non-compact group, let K be the compact subgroup fixed by a Cartan involution and assume G/K is an exceptional, symmetric space, one of Cartan type E,F or G. We find the minimal integer, L(G), such that any convolution product of L(G) continuous, K-bi-invariant measures on G is absolutely continuous with respect to Haar measure. Further, any product of L(G) double cosets has non-empty interior. The number L(G) is either 2 or 3, depending on the Cartan type, and in most cases is strictly less than the rank of G. 1. Introduction In [18] Ragozin showed that if G is a connected simple Lie group and K a compact, connected subgroup fixed by a Cartan involution of G, then the convo- lution of any dim G/K, continuous, K-bi-invariant measures on G is absolutely continuous with respect to the Haar measure on G. This was improved by Graczyk and Sawyer in [8], who showed that when G is non-compact and n = rank G/K, then the convolution product of any n + 1 such measures is absolutely continuous. They conjectured that n + 1 was sharp with this property. This conjecture was shown to be false, in general, for the symmetric spaces of classical Cartan type in [11] (where it was shown that rank G/K suffices except when the restricted root system is type An). -
The Absolute Continuity of Convolutions of Orbital Measures in Symmetric Spaces
THE ABSOLUTE CONTINUITY OF CONVOLUTIONS OF ORBITAL MEASURES IN SYMMETRIC SPACES SANJIV KUMAR GUPTA AND KATHRYN E. HARE Abstract. We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characteri- zation can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rankG/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rankG/K +1 is needed. 1. Introduction In this paper, we study the smoothness properties of K-bi-invariant measures on the irreducible Riemannian symmetric spaces G/K, where G is a connected Lie group and K is a compact, connected subgroup fixed by a Cartan involution of G. Inspired by earlier work of Dunkl [6] on zonal measures on spheres, Ragozin in [27] and [28] showed that the convolution of any dim G/K, continuous, K-bi-invariant measures on G was absolutely continuous with respect to the Haar measure on G. This was improved by Graczyk and Sawyer, who in [10] showed that if G was non-compact and n = rankG/K, then any n + 1 convolutions of such measures is absolutely continuous and that this is sharp for the symmetric spaces whose restricted root system was type An.