DOCSLIB.ORG

On Suslin Matrices and Their Connection to Spin Groups Vineeth Chintala

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Suslin Matrices and Their Connection to Spin Groups Vineeth Chintala On Suslin Matrices and their connection to Spin groups Vineeth Chintala Introduction William Clifford (1878) constructed his algebras in order to gen- eralise Complex numbers and Quaternions to higher dimensions. Though Clifford algebras have found wide applications in geometry, quantum mechanics and applied mathematics, their abstract con- struction may not be easy to work with. Even in the simplest case for a hyperbolic quadratic form, where Clifford algebras are total matrix rings, there is no simple isomorphism describing their generators. In this paper, we will give a simple description of Clifford algebras us- ing what are called Suslin matrices. The identities followed by these matrices will then be used to study the corresponding Spin groups. Conversely one will now be able to relate some (seemingly) acciden- tal properties of Suslin matrices to the geometry of Clifford algebras. Let R be a commutative ring and V be a free R-module. This ar- ticle is on the Clifford algebra of the hyperbolic module H(V) = V ⊕ V∗ equipped with the quadratic form q(x, f ) = f (x). These algebras are isomorphic to total matrix rings of size 2n × 2n, where dim V = n. The representation of Clifford algebras in terms of Suslin matrices is faithful only when dim V is odd and we will mostly be concerned with this case. With an explicit construction at hand, one can expect to derive some results on the Clifford algebra and its Spin group us- ing simple matrix computations. For example, the identities followed by the Suslin matrices (in Equation 1) give us (as we will see) the involution on the Clifford algebra. Indeed given any element of the Clifford algebra we will be able to explicitly construct its conjugate matrix under the involution. As an application of Suslin matrices, we give a proof of the following exceptional isomorphisms : arXiv:2009.12107v2 [math.RA] 16 Dec 2020 ∼ ∼ Spin4(R) = SL2(R) × SL2(R), Spin6(R) = SL4(R). On the other hand, some key properties of Suslin matrices (see Theorem 2.4 and the margin-notes in 7.3) are re-derived here (with minimal computation) using the connection to Clifford algebras. These properties play the crucial role in the papers [JR], [JR2] which study an action of the orthogonal group on unimodular rows. In addition, this connection to Clifford algebras explains why the size of a Suslin matrix has to be of the form 2n−1 × 2n−1. One can go on on suslin matrices and their connection to spin groups 2 and ask if there are other matrix-constructions, possibly of smaller sizes. Indeed, it turns out that the size chosen by Suslin is the least possible. This is an embedding problem of a quadratic space into the algebra of matrices - it belongs to a larger theme of finding natural ways of constructing algebras out of quadratic spaces. It would be interesting to see similar constructions for other types of quadratic forms - and for this reason alone, it might be worthwhile having this connection between Suslin matrices and Clifford algebras. We begin with a few preliminaries on Clifford algebras and Suslin matrices (to make the article accessible) and the link between them. The involution of the Clifford algebra is then described via some identities followed by the Suslin matrices. It turns out that the behav- ior of the Suslin matrices (hence that of the Spin groups) depends on whether the dimension of V is odd or even. One realizes this after observing the pattern followed by the involution in the odd and even cases. This leads us to Section 5 where the action of the Spin group on the Suslin space is studied in detail. The last part of the paper contains a few remarks on the Spin and Epin groups. By a ring, we always mean a ring with unity. In this paper, V will always denote a free R-module where R is any commutative ring and H(V) = V ⊕ V∗. From now on, we fix the standard basis for V = Rn and identify V with its dual V∗. One can then write the quadratic form on H(V) as T q(v, w) = v · w = a1b1 + ··· + anbn. 1 1 for v = (a1, ··· , an), w = (b1, ··· , bn). References : For general literature on Clifford algebras and Spin groups Acknowledgements. I would like to thank my advisor Prof. Ravi over a commutative ring, the reader is referred to [B1], [B2] of H. Bass. Rao for the many discussions we had while working on this paper. For a more detailed introduction to His willingness to give his time so generously has been very much Suslin matrices (and their important appreciated. connection to unimodular rows), a few references are Suslin’s paper [S], the Part of this work was done during a short enjoyable visit to papers [JR], [JR2] of Ravi Rao and Selby Fields Institute (in Toronto), which funded me to attend its Spring Jose, and T. Y. Lam’s book [L] (sections III.7, VIII.5). school on Torsors in 2013. 1 The Suslin Construction The Suslin construction gives a sequence of matrices whose size doubles at every step. Moreover each Suslin matrix S has a conjugate Suslin matrix S such that SS and S + S are scalars (norm and the trace). Let us pause here to see the recursive process by which the Suslin on suslin matrices and their connection to spin groups 3 n n T 2n−1 matrix Sn(v, w) of size 2 × 2 , and determinant (v · w ) , is con- structed from two vectors v, w in Rn+1. n Let v = (a0, v1), w = (b0, w1) where v1, w1 are vectors in R . Define ! a0 v1 S0(a0, b0) = a0, S1(v, w) = −w1 b0 and 0 1 a0 I2n−1 Sn−1(v1, w1) B C Sn(v, w) = @ A . T −Sn−1(w1, v1) b0 I2n−1 For Sn = Sn(v, w), define 0 1 b0 I2n−1 −Sn−1(v1, w1) T B C Sn := Sn(w, v) = @ A . T Sn−1(w1, v1) a0 I2n−1 For example if v = (a1, a2) and w = (b1, b2), then ! ! a a b −a S(v, w) = 1 2 , S(v, w) = 1 2 . −b2 b1 b2 a1 For v = (a1, a2, a3) and w = (b1, b2, b3), 0 1 0 1 a1 0 a2 a3 b1 0 −a2 −a3 B − C B − C B 0 a1 b3 b2C B 0 b1 b3 b2C S(v, w) = B C , S(v, w) = B C . @−b2 a3 b1 0 A @b2 −a3 a1 0 A −b3 −a2 0 b1 b3 a2 0 a1 For v = (a1, a2, a3, a4) and w = (b1, b2, b3, b4), the corresponding 8 × 8 Suslin matrices are 0 1 a1 0 0 0 a2 0 a3 a4 B C B 0 a1 0 0 0 a2 −b4 b3C B C B 0 0 a1 0 −b3 a4 b2 0 C B C B 0 0 0 a −b −a 0 b C B 1 4 3 2C B C S(v, w) = B C B C B−b2 0 a3 a4 b1 0 0 0 C B C B 0 −b −b b 0 b 0 0 C B 2 4 3 1 C B C @−b3 a4 −a2 0 0 0 b1 0 A −b4 −a3 0 −a2 0 0 0 b1 on suslin matrices and their connection to spin groups 4 and 0 1 b1 0 0 0 −a2 0 −a3 −a4 B C B 0 b1 0 0 0 −a2 b4 −b3C B C B 0 0 b1 0 b3 −a4 −b2 0 C B C B 0 0 0 b b a 0 −b C B 1 4 3 2C B C S(v, w) = B C . B C Bb2 0 −a3 −a4 a1 0 0 0 C B C B 0 b b −b 0 a 0 0 C B 2 4 3 1 C B C @b3 −a4 a2 0 0 0 a1 0 A b4 a3 0 a2 0 0 0 a1 Notice that Sn(v, w) is also a Suslin matrix. Indeed, Sn(v, w) = 0 0 Sn(v , w ) where 0 ! ! v (b0, −v1) 0 = . w (a0, −w1) In particular, we have S0(a0, b0) = b0. One can easily check that a Suslin matrix Sn = Sn(v, w) satisfies the following properties: T SnSn = SnSn = (v · w )I2n , T 2n−1 det Sn = (v · w ) . Unless otherwise specified, we assume that n > 0. Then the element (v, w) is determined by its corresponding Suslin matrix Sn(v, w). So we sometimes identify the element (v, w) with Sn(v, w) for n > 0. The set of Suslin matrices of size 2n × 2n is an R-module under matrix-addition and scalar multiplication given by rSn(v, w) = Sn(rv, rw) for r 2 R; Moreover when n > 0, the mapping Sn ! Sn is an isometry of the quadratic space H(Rn+1), preserving the quadratic form q(v, w) = v · wT. In his paper [S], A. Suslin then describes a sequence of matrices Jn 2 M2n (R) by the recurrence formula 8 >1 for n = 0 > > > >0 1 > > Jn−1 0 <>@ A for n even Jn = 0 −Jn−1 > > > >0 1 > 0 Jn−1 > for n odd. >@ A : −Jn−1 0 It is easy to check that det(J) = 1 and 2 2 We will simply write J and S (or S(v, w)) and drop the subscript when there is no confusion. When there is some confusion, remember that the subscript r in Sr is used to indicate that r r Sr (or Jr) is a 2 × 2 matrix.
Load more

Recommended publications
  • The Unitary Representations of the Poincaré Group in Any Spacetime
    The unitary representations of the Poincar´e group in any spacetime dimension Xavier Bekaert a and Nicolas Boulanger b a Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´ed’Orl´eans, CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] b Service de Physique de l’Univers, Champs et Gravitation Universit´ede Mons – UMONS, Place du Parc 20, 7000 Mons (Belgium) [email protected] An extensive group-theoretical treatment of linear relativistic field equa- tions on Minkowski spacetime of arbitrary dimension D > 2 is presented in these lecture notes. To start with, the one-to-one correspondence be- tween linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representa- tions reduces the problem of classifying the representations of the Poincar´e group ISO(D 1, 1) to the classification of the representations of the sta- − bility subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D 4 <n<D ). Finally, covariant field equations are given for each − unitary irreducible representation of the Poincar´egroup with non-negative arXiv:hep-th/0611263v2 13 Jun 2021 mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams.
    [Show full text]
  • 14.2 Covers of Symmetric and Alternating Groups
    Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates.
    [Show full text]
  • IMAGE of FUNCTORIALITY for GENERAL SPIN GROUPS 11 Splits As Two Places in K Or V Is Inert, I.E., There Is a Single Place in K Above the Place V in K
    IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS MAHDI ASGARI AND FREYDOON SHAHIDI Abstract. We give a complete description of the image of the endoscopic functorial transfer of generic automorphic representations from the quasi-split general spin groups to general linear groups over arbitrary number fields. This result is not covered by the recent work of Arthur on endoscopic classification of automorphic representations of classical groups. The image is expected to be the same for the whole tempered spectrum, whether generic or not, once the transfer for all tempered representations is proved. We give a number of applications including estimates toward the Ramanujan conjecture for the groups involved and the characterization of automorphic representations of GL(6) which are exterior square transfers from GL(4); among others. More applications to reducibility questions for the local induced representations of p-adic groups will also follow. 1. Introduction The purpose of this article is to completely determine the image of the transfer of globally generic, automorphic representations from the quasi-split general spin groups to the general linear groups. We prove the image is an isobaric, automorphic representation of a general linear group. Moreover, we prove that the isobaric summands of this representation have the property that their twisted symmetric square or twisted exterior square L-function has a pole at s = 1; depending on whether the transfer is from an even or an odd general spin group. We are also able to determine whether the image is a transfer from a split group, or a quasi-split, non-split group. Arthur's recent book on endoscopic classification of representations of classical groups [Ar] establishes similar results for automorphic representations of orthogonal and symplectic groups, generic or not, but does not cover the case of the general spin groups we consider here.
    [Show full text]
  • 1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
    1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5).
    [Show full text]
  • 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).
    [Show full text]
  • Maximal Abelian Subgroups of Spin Groups and Some Exceptional
    MAXIMAL ABELIAN SUBGROUPS OF SPIN GROUPS AND SOME EXCEPTIONAL SIMPLE LIE GROUPS JUN YU Abstract. We classify closed abelian subgroups of the simple groups G2, F4, Aut(so(8)) having centralizer the same dimension as the dimension of the sub- group, as well as finite abelian subgroups of certain spin and half-spin groups having finite centralizer. Mathematics Subject Classification (2010). 20E07, 20E45, 20K27. Keywords. Maximal abelian subgroup, Weyl group. Contents 1. Introduction 1 2. Spin and half-spin groups 3 3. Compact simple Lie groups G2 and F4 11 3.1. Preliminaries 11 3.2. Compact simple Lie group of type G2 12 3.3. Compact simple Lie group of type F4 12 4. Compact simple Lie groups of type D4 14 4.1. Preliminaries 14 4.2. Outer automorphisms of order three 15 4.3. Orthogonal group 17 4.4. Finite abelian subgroups 19 References 21 arXiv:1403.2679v1 [math.GR] 11 Mar 2014 1. Introduction In this paper we continue the classification of maximal abelian subgroups of com- pact simple Lie groups in [Yu2], where we give a classification of such subgroups of matrix groups. The same as in [Yu2], we study closed abelian subgroups F of a compact simple Lie group G satisfying the condition of ( ) dim gF = dim F, ∗ 0 where g0 is the Lie algebra of G. Precisely, we discuss such subgroups of compact simple Lie groups G2, F4, and of the automorphism group Aut(so(8)) of so(8), as well as finite abelian subgroups of the spin group Spin(n) for n = 7, 8, 10, 12, 14, and of the half-spin group Spin(12)/ c .
    [Show full text]
  • Essential Dimension of the Spin Groups in Characteristic 2
    Essential dimension of the spin groups in characteristic 2 Burt Totaro The essential dimension of an algebraic group G is a measure of the number of parameters needed to describe all G-torsors over all fields. A major achievement of the subject was the calculation of the essential dimension of the spin groups over a field of characteristic not 2, started by Brosnan, Reichstein, and Vistoli, and completed by Chernousov, Merkurjev, Garibaldi, and Guralnick [3, 4, 7], [17, Theorem 9.1]. In this paper, we determine the essential dimension of the spin group Spin(n) for n ≥ 15 over an arbitrary field (Theorem 2.1). We find that the answer is the same in all characteristics. In contrast, for the groups O(n) and SO(n), the essential dimension is smaller in characteristic 2, by Babic and Chernousov [1]. In characteristic not 2, the computation of essential dimension can be phrased to use a natural finite subgroup of Spin(2r + 1), namely an extraspecial 2-group, a central extension of (Z/2)2r by Z/2. A distinctive feature of the argument in characteristic 2 is that the analogous subgroup is a finite group scheme, a central r r extension of (Z/2) × (µ2) by µ2, where µ2 is the group scheme of square roots of unity. In characteristic not 2, Rost and Garibaldi computed the essential dimension of Spin(n) for n ≤ 14 [6, Table 23B], where case-by-case arguments seem to be needed. We show in Theorem 3.1 that for n ≤ 10, the essential dimension of Spin(n) is the same in characteristic 2 as in characteristic not 2.
    [Show full text]
  • Representations of Lorentz and Poincaré Groups
    Representations of Lorentz and Poincar´egroups Joseph Maciejko CMITP and Department of Physics, Stanford University, Stanford, California, 94305 I. LORENTZ GROUP We consider first the Lorentz group O(1, 3) with infinitesimal generators J µν and the associated Lie algebra given by [J µν,J ρσ] = i(gνρJ µσ − gµρJ νσ − gνσJ µρ + gµσJ νρ) (1) In quantum field theory, we actually consider a subgroup of O(1, 3), the proper orthochronous + 0 or restricted Lorentz group SO (1, 3) = {Λ ∈ O(1, 3)| det Λ = 1 and Λ 0 ≥ 0}, which ex- cludes parity and time-reversal transformations that are thus considered as separate, discrete operations P and T . A generic element Λ of the Lorentz group is given by exponentiating 1 µν the generators together with the parameters of the transformation, Λ = exp(−iωµνJ /2). The Lorentz group has both finite-dimensional and infinite-dimensional representations. However, it is non-compact, therefore its finite-dimensional representations are not unitary (the generators are not Hermitian). The generators of the infinite-dimensional representa- tions can be chosen to be Hermitian. A. Finite-dimensional representations We first study the finite-dimensional representations of SO+(1, 3). These representations act on finite-dimensional vector spaces (the base space). Elements of these vector spaces are said to transform according to the given representation. Trivial representation. In the trivial representation, we have the one-dimensional representation J µν = 0. Hence any Lorentz transformation Λ is represented by 1. This representation acts on a one-dimensional vector space whose elements are 1-component ob- jects called Lorentz scalars.
    [Show full text]
  • Essential Dimension of the Spin Groups in Characteristic 2
    Essential dimension of the spin groups in characteristic 2 Burt Totaro The essential dimension of an algebraic group G is a measure of the number of parameters needed to describe all G-torsors over all fields. A major achievement of the subject was the calculation of the essential dimension of the spin groups over a field of characteristic not 2, started by Brosnan, Reichstein, and Vistoli, and completed by Chernousov, Merkurjev, Garibaldi, and Guralnick [3, 4, 7], [18, Theorem 9.1]. In this paper, we determine the essential dimension of the spin group Spin(n) for n ≥ 15 over an arbitrary field (Theorem 2.1). We find that the answer is the same in all characteristics. In contrast, for the groups O(n) and SO(n), the essential dimension is smaller in characteristic 2, by Babic and Chernousov [1]. In characteristic not 2, the computation of essential dimension can be phrased to use a natural finite subgroup of Spin(2r + 1), namely an extraspecial 2-group, a central extension of (Z=2)2r by Z=2. A distinctive feature of the argument in characteristic 2 is that the analogous subgroup is a finite group scheme, a central r r extension of (Z=2) × (µ2) by µ2, where µ2 is the group scheme of square roots of unity. In characteristic not 2, Rost and Garibaldi computed the essential dimension of Spin(n) for n ≤ 14 [6, Table 23B], where case-by-case arguments seem to be needed. We show in Theorem 4.1 that for n ≤ 10, the essential dimension of Spin(n) is the same in characteristic 2 as in characteristic not 2.
    [Show full text]
  • Local Descent to Quasi-Split Even General Spin Groups
    Local descent to quasi-split even general spin groups Eyal Kaplan∗1, Jing Feng Lau2, and Baiying Liu†3 1Department of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel, [email protected] 2Singapore University of Social Sciences, 463 Clementi Road, Singapore 599494, Singapore, jfl[email protected] 3Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, U.S.A., [email protected] March 14, 2019 Abstract Let n ¥ 2 and τ be an irreducible unitary supercuspidal representation of GLp2nq over a local non-archimedean field. Assuming the twisted symmetric square L-function of τ has a pole at s “ 0, we construct the local descent of τ to the corresponding general spin group of even rank (split over the base field, or over a quadratic extension). We show that this local descent is non-trivial, generic, unitary and supercuspidal. Moreover, any generic irreducible supercuspidal representation of the general spin group which lifts functorially to τ is contragradient to some constituent of the representation we construct. Keywords: Local Descent, GSpin Groups, Local Langlands Functoriality. 2000 Mathematics Subject Classification Primary 11F70, 22E50; Secondary 11F85, 22E55. 1 Introduction Let G be a connected reductive algebraic group defined over a global field K or local field F , let G be the group of rational points over that field. One of the central problems in the Langlands program is to determine the functorial image of a Langlands functorial lift corresponding to some L- homomorphism from LG to a suitable general linear group. To this end, when G is a classical group, with the exception of some low rank cases, it has been shown that the local and global descent method is an effective approach in proving that the relevant functorial image is contained in the collection of irreducible admissible representations of certain p-adic linear groups or automorphic representations of adelic linear groups satisfying a list of conditions, cf.
    [Show full text]
  • Lecture 3: Spinor Representations
    Spin 2010 (jmf) 19 Lecture 3: Spinor representations Yes now I’ve met me another spinor... — Suzanne Vega (with apologies) It was Élie Cartan, in his study of representations of simple Lie algebras, who came across repres- entations of the orthogonal Lie algebra which were not tensorial; that is, not contained in any tensor product of the fundamental (vector) representation. These are the so-called spinorial representations. His description [Car38] of the spinorial representations was quite complicated (“fantastic” according to Dieudonné’s review of Chevalley’s book below) and it was Brauer and Weyl [BW35] who in 1935 de- scribed these representations in terms of Clifford algebras. This point of view was further explored in Chevalley’s book [Che54] which is close to the modern treatment. This lecture is devoted to the Pin and Spin groups and to a discussion of their (s)pinorial representations. 3.1 The orthogonal group and its Lie algebra Throughout this lecture we will let (V,Q) be a real finite-dimensional quadratic vector space with Q nondegenerate. We will drop explicit mention of Q, whence the Clifford algebra shall be denoted C(V) and similarly for other objects which depend on Q. We will let B denote the bilinear form defining Q. We start by defining the group O(V) of orthogonal transformations of V: (51) O(V) {a :V V Q(av) Q(v) v V} . = → | = ∀ ∈ We write O(Rs,t ) O(s,t) and O(n) for O(n,0). If a O(V), then deta 1. Those a O(V) with deta = ∈ =± ∈ = 1 define the special orthogonal group SO(V).
    [Show full text]
  • 5. SPINORS 5.1. Prologue. 5.2. Clifford Algebras and Their
    5. SPINORS 5.1. Prologue. 5.2. Clifford algebras and their representations. 5.3. Spin groups and spin representations. 5.4. Reality of spin modules. 5.5. Pairings and morphisms. 5.6. Image of the real spin group in the complex spin module. 5.7. Appendix: Some properties of the orthogonal groups. 5.1. Prologue. E. Cartan classified simple Lie algebras over C in his thesis in 1894, a classification that is nowadays done through the (Dynkin) diagrams. In 1913 he classified the irreducible finite dimensional representations of these algebras1. For any simple Lie algebra g Cartan's construction yields an irreducible representation canonically associated to each node of its diagram. These are the so-called fun- damental representations in terms of which all irreducible representations of g can be constructed using and subrepresentations. Indeed, if πj(1 j `) are the ⊗ ≤ ≤ fundamental representations and mj are integers 0, and if vj is the highest vector ≥ of πj, then the subrepresentation of m1 m` π = π⊗ ::: π⊗ 1 ⊗ ⊗ ` generated by m1 m` v = v⊗ ::: v⊗ 1 ⊗ ⊗ ` is irreducible with highest vector v, and every irreducible module is obtained in this manner uniquely. As is well-known, Harish-Chandra and Chevalley (independently) developed around 1950 a general method for obtaining the irreducible representa- tions without relying on case by case considerations as Cartan did. `+1 j If g = sl(` + 1) and V = C , then the fundamental module πj is Λ (V ), and all irreducible modules can be obtained by decomposing the tensor algebra over the 1 defining representation V . Similarly, for the symplectic Lie algebras, the decomposi- tion of the tensors over the defining representation gives all the irreducible modules.
    [Show full text]