This is page 271 Printer: Opaque this 17 Models for Irreducible Representations of Spin(m) P. Van Lancker, F. Sommen∗ and D. Constales ABSTRACT In this paper we consider harmonic and monogenic polynomials of simplicial type. It is proved that these polynomials provide explicit realizations of all irreducible representations of Spin(m). 1 Introduction Let (e1 , . . . , em ) be an orthonormal basis of Euclidean space Rm endowed Pm m with the inner product hx, yi = i=1 xi yi , x, y ∈ R . By Rm (Cm ) we m denote the real (complex) 2 -dimensional Clifford algebra over Rm generated by the relations e2i = −1, i =P 1, . . . , m and ei ej + ej ei = 0, i 6= j. An element of Cm is of the form a = A⊂M aA eA , aA ∈ C, M = {1, . . . , m} and eφ = e0 = 1. Reversion on Cm is the (principal) anti-involution des(s−1) fined by ẽA = (−1) 2 , s = ♯A and extended by linearity to Cm . Con∗ Senior Research Associate, FWO, Univ. Gent, Belgium 1991 Mathematics Subject Classification. 30G35, 58G20. Keywords: Clifford analysis, Dirac operators, Representations, Spin groups. Advances in Applied Clifford Algebras 11 (S1) 271-289, 2001 c 2001 Universidad Nacional Autónoma de México. Printed in Mexico 272 Models for Irreducible Representations of Spin(m). P jugation on Cm is the anti-involution on Cm given by ā = A⊂M āA ēA = 1, . . . , m. Vectors x ∈ Rm where ēA = ēαh . . . ēα1 and ēj = −ej , j P m are identified with Clifford numbers x = j=1 xj ej . The following subgroups of the real Clifford algebra Rm are of interest. The Pin group P in(m) is the group consisting of products of unit vectors in Rm ; the Spin group Spin(m) is the subgroup of P in(m) consisting of products of an even number of unit vectors in Rm . For an element s ∈ P in(m) the map χ(s) : Rm → Rm : x 7→ sxs̃ induces an orthogonal transformation of Rm . In this way P in(m) defines a double covering of the orthogonal group O(m). The restriction of this map to Spin(m), x :7→ sxs̄ then defines a double covering of the rotation group SO(m). The Dirac operator on Rm is given by ∂x = e1 ∂x1 + . . . + em ∂xm . In spherical coordinates x = ρω, ρ = |x| = (x21 + . . . + x2m )1/2 and ω ∈ S m−1 , S m−1 being the unit sphere in Rm , the Dirac operator admits the polar decomposition ∂x = ω(∂ρ + ρ1 Γω ) where Γω = −x ∧ ∂x is the spherical Dirac operator on S m−1 . In terms of the momentum operators Lij P = xi ∂xj − xj ∂xj , i, j = 1, . . . , m on Rm the Γ-operator is given by Γ = − i<j eij Lij while P the Laplace-Beltrami operator △S = i<j L2ij = Γ(m − 2 − Γ). The theory of harmonic functions of a matrix variable was presented in detail in [GM]. They consider simplicial harmonics (i.e. harmonic polynomials of a matrix variable invariant under the action of SL(r)) which provide models for irreducible representations of SO(m) with integer weight (k, . . . , k, 0, . . . , 0) (r times k). This leads to the idea to look for models of half integer weight irreducible representations of Spin(m) inside spaces of monogenic functions of several vector variables. This theory was already developed to some extent in [Co] (in the case of several quaternionic variables see e.g. [ABLSS], [Pa] and [Pe]). As a matter of fact, to obtain polynomial irreducible representations of Spin(m) we look to spaces of polynomials which are already irreducible with respect to the action of GL(m). These are the so called simplicial polynomials or polynomials of Young type. To obtain models for all integer (half integer) weight representations we then impose harmonicity (monogenicity) conditions. This leads to the notion of simplicial harmonic (monogenic) system. The models for the irreducible representations of Spin(m) arise from the construction of specific highest weight vectors. In the framework of Clifford algebra, weight vectors for the fundamental representations were first constructed in [DS]. Later on weight vectors for arbitrary (half)-integer weights were given in [So3]. Although these weight vectors satisfy the simplicial (monogenic) harmonic system, it took some extra ideas by the first author and basic facts from [FH] to prove that they generate the spaces of simplicial (spinor valued monogenic) scalar valued harmonic polynomials. As a result the simplicial harmonic and monogenic system are (up to isomorphism) the most refined Spin(m)-invariant systems of partial differential equations and thus provide the basic building blocks for any Spin(m)-invariant system. P. Van Lancker, F. Sommen, D. Constales 273 2 Irreducible Representations of GL(m) and Polynomials of Simplicial Variables Pm Polynomials of k vector variables x1 , . . . , xk where xl = j=1 xlj ej can be regarded as polynomials on Rk×m or on Rkm by the identification:     x11 · · · x1m x1    ..  = (x ) X =  ...  =  ... lj .  xk xk1 ··· xkm The space of these polynomials will be denoted as P[x1 , . . . , xk ]. Its subspace of polynomials homogeneous of degree li in each vector variable xi will be denoted by Pl1 ,...,lk [x1 , . . . , xk ]. The Fischer inner product on these space is the usual Fischer inner product on polynomials of km scalar variables given by   ∂ ∂ P̄ ( hP (x1 , . . . , xk ), Q(x1 , . . . , xk )i = )Q(x1 , . . . , xk ) (0) ,..., ∂x1 ∂xk 0   ∂ )Q(xlj ) (0) P̄ ( = ∂xlj 0 and hR((g t )−1 )P, R(g)Qi = hP, Qi. Obviously polynomials of different degree of homogeneity are orthogonal with respect to this inner product. The right regular representation of GL(m) on P[x1 , . . . , xm ] or on a subspace of homogeneous polynomials of fixed degree of homogeneity is given by: R(g)P (x1 , . . . , xm ) = P (Xg) = P (x1 g, . . . , xm g), g ∈ GL(m). Up to equivalence all irreducible representations of GL(m) can be labelled by m-tuples l = (l1 , . . . , lm ) of integers such that l1 ≥ · · · ≥ lm . An explicit realization of these irreducible representations within the space Pl1 ,...,lm [x1 , . . . , xm ] can be obtained by imposing row homogeneity conditions on these polynomials. This can be achieved by specifying an extra left group action on Pl1 ,...,lm [x1 , . . . , xm ]. This goes as follows (see e.g. [GR]). Let Nm ⊂ GL(m) be the subgroup of GL(m) consisting of lower triangular matrices such that all elements on the diagonal are one. The subspace of Pl1 ,...,lm [x1 , . . . , xm ] consisting of polynomials invariant under the left acm tion of Nm , i.e. P (Nm X) = P (X) is denoted by PlN1 ,...,l [x1 , . . . , xm ]. It m can be proved that this space is irreducible for the right regular representation of GL(m) and provides a model for the irreducible representation with weight l = (l1 , . . . , lm ). We write: m PlN1 ,...,l [x1 , . . . , xm ] ∼ = (l1 , . . . , lm ). m 274 Models for Irreducible Representations of Spin(m). This irreducible representation is completely determined by specifying its highest weight vector wl1 ,...,lm (x1 , . . . , xm ; e1 , . . . , em ) = hx1 e1 il1 −l2 hx1 ∧ x2 e1 ∧ e2 il2 −l3 · · · hx1 ∧ · · · ∧ xm e1 ∧ · · · ∧ em ilm where hx1 ∧ · · · ∧ xk e1 ∧ · · · ∧ ek i = −[(x1 ∧ · · · ∧ xk )(e1 ∧ · · · ∧ e k )]0 hx1 e1 i · · · hx1 ek i   .. .. = det  . . . hxk e1 i Thus ··· hxk ek i m PlN1 ,...,l [x1 , . . . , xm ] = spanR {R(g)wl1 ,...,lm }. m It follows from Schur’s lemma that the weight vector wl1 ,...,lm is the reprom ducing kernel of PlN1 ,...,l [x1 , . . . , xm ], i.e. m m m PlN (·)i (x1 , . . . , xm ) = Dl1 ,...,lm hw̄l1 ,...,lm (x1 , . . . , xm ; ·), PlN 1 ,...,lm 1 ,...,lm for some non zero constant Dl1 ,...,lm . Consider now the representation of the t upper triangular subgroup (all diagonal elements equal to one) Um = Nm on Pl1 ,...,lm [x1 , . . . , xm ] given by: ρ(u)P (X) = P (ut X), u ∈ Um and its derived representation given by: (ρ̃(A)P )(X) = d ((ρ(exp tA)P )(X)) |t=0 dt where A belongs to the Lie algebra of Um . This algebra can be identified with the algebra generated by the vector fields hxi ∂xj i; j > i or equivalently by the algebra generated by hxi ∂xi+1 i; i = 1, . . . , m − 1. Summarizing we thus get the following equivalent characterizations of the irreducible representation l = (l1 , . . . , lm ). m PlN1 ,...,l [x1 , . . . , xm ] m = {P ∈ Pl1 ,...,lm [x1 , . . . , xm ] : hxi ∂xi+1 iP = 0, i = 1, . . . , m − 1} = {Pl1 ,...,lm (x1 , . . . , xm ) = Pl1 ,...,lm (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xm )} A pure k-vector of the form x1 ∧· · ·∧xk is called a simplicial variable and a variable of the form x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xm is called a flag variable. A polynomial P (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xm ) depending on a flag variable will be referred to as a simplicial polynomial. P. Van Lancker, F. Sommen, D. Constales 275 3 Simplicial Harmonic and Monogenic Polynomials In this section we define some important SO(m) and Spin(m) invariant systems of partial differential equations. Let P ∈ P[x1 , . . . , xk ]. Then we call P harmonic if P satisfies the harmonic system of equations: △xi P (x1 , . . . , xk ) = 0, i = 1, . . . , k h∂xi ∂xj iP (x1 , . . . , xk ) = 0, i 6= j where △xi denotes the Laplacian in the vector variable xi . The space of these polynomials will be denoted by H[x1 , . . . , xk ]. This definition of harmonicity corresponds to the notion of harmonic polynomials of matrix variable described by Gilbert and Murray in [GM]. A polynomial P is called monogenic in several vector variables if it satisfies the monogenic system (see also [Co], [Pe]): ∂xi P (x1 , . . . , xk ) = 0, i = 1, . . . , k. (1) The space of monogenic polynomials is denoted by M[x1 , . . . , xk ]. Clearly the monogenic system refines the harmonic system: M[x1 , . . . , xk ] ⊂ H[x1 , . . . , xk ]. The corresponding Fischer decompositions are given by (see also [GM], [Co] and [So1]): P[x1 , . . . , xk ] = = H[x1 , . . . , xk ] ⊕⊥ = M[x1 , . . . , xk ] ⊕⊥ k X |xi |2 P[x1 , . . . , xk ] + i=1 k X i=1 X i<j ! xi P[x1 , . . . , xk ] .  hxi xj iP[x1 , . . . , xk ] It is important to notice that the decompositions between brackets are not unique. Only the harmonic or monogenic part of a polynomial are uniquely determined. These systems can be further refined by considering them on homogeneous polynomials of simplicial type, leading to the following definitions. A polynomial P ∈ Pl1 ,...,lk [x1 , . . . , xk ] satisfies the simplicial harmonic system if P is harmonic and of simplicial type, i.e: △xi P (x1 , . . . , xk ) = 0, i = 1, . . . , k h∂xi ∂xj iP (x1 , . . . , xk ) = 0, i 6= j hxi ∂xi+1 iP (x1 , . . . , xk ) = 0, i = 1, . . . , k − 1. The Lie algebra generated by the operators △1 , hxi , ∂xi+1 i, i = 1, . . . , k − 1 is the algebra consisting of the operators determining the harmonic system, 276 Models for Irreducible Representations of Spin(m). together with the operators coming from the action of the upper triangular group Uk . Hence the simplicial harmonic system is equivalent to x1 7→ Pl1 ,...,lk (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xk ) (2) is harmonic in x1 . This space will be denoted by HlN1 k,...,lk [x1 , . . . , xk ]. Simplicial harmonic polynomials of the form P (x1 ∧ · · · ∧ xk ) are exactly the harmonics studied by Gilbert and Murray in connection with equal weight representations of SO(m). A polynomial P ∈ Pl1 ,...,lk [x1 , . . . , xk ] satisfies the simplicial monogenic system if P is monogenic and of simplicial type, i.e: ∂xi P (x1 , . . . , xk ) = 0, i = 1, . . . , k hxi ∂xi+1 iP (x1 , . . . , xk ) = 0, i = 1, . . . , k − 1, or equivalently, by taking commutators of ∂x1 , hxi , ∂xi+1 i x1 7→ Pl1 ,...,lk (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xk ) (3) k is monogenic x1 . This space will be denoted by MN l1 ,...,lk [x1 , . . . , xk ]. 4 Irreducible Representations of Spin(m) Let us recall some facts related to the algebraic construction of irreducible representations of Spin(m) (see also [GM] and [FH]). Up to equivalence the unitary irreducible Spin(m)-modules can be labelled by considering the action of the maximal torus of Spin(m): 1 m 1 T = {s = exp( e12 t1 ) · · · exp( e2M −1,2M tM ), tj ∈ R, M = [ ]}. 2 2 2 Let R(s) : Spin(m) → V be an irreducible representation of Spin(m). If we restrict this representation to the maximal abelian subgroup T of Spin(m), the space V splits into weight subspaces generated by eigenvectors v satisfying 1 1 R(exp( e12 t1 ) · · · exp( e2M −1,2M tM ))v = exp i(l1 t1 + · · · + lM tM )v. 2 2 The eigenvalues are determined by M -tuples l = (l1 , . . . , lM ) consisting entirely of either integer or half integer numbers. They are the so called weights of the representation. These weights can be ordered lexicograph′ ically: l = (l1 , . . . , lM ) > l′ = (l1′ , . . . , lM ) if the first non zero difference li − li′ is positive. In this way V may be identified with an ordered set of M -tuples which is the same for equivalent representations. In this set of M tuples a unique weight can be singled out by considering the action of the P. Van Lancker, F. Sommen, D. Constales 277 Weyl group on the ordered weights. The Weyl group acts as a permutation group on the numbers determining the weights together with an arbitrary or even number of changes of signs when m is odd or even. Factoring out this action one can see that V contains a unique highest weight with respect to the ordering defined above. These are called highest weights and are of the form: l = (l1 , . . . , lM ) l = (l1 , . . . , lM ) : l1 ≥ l2 ≥ . . . ≥ lM if m = 2M + 1 : l1 ≥ l2 ≥ . . . ≥ |lM | if m = 2M where all li ∈ Z or all li ∈ 12 Z. By a theorem of Cartan the weight subspace corresponding to the highest weight is one dimensional; it is generated by the highest weight vector (defined up to a multiple). Moreover each M -tuple of the form above is actually the highest weight of exactly one irreducible representation of Spin(m). This gives the correspondence between highest weights or highest weight vectors and unitary irreducible Spin(m)-modules. Of particular importance are the so called fundamental (the notion of fundamental we use is not the standard one) weights. These are the highest weights of the form: 1 1 (1, 0, . . . , 0), . . . , (1, . . . , 1), ( , . . . , ) 2 2 if m = 2M + 1, and 1 1 1 1 1 (1, 0, . . . , 0), . . . , (1, . . . , 1), ( , . . . , ), (1, . . . , 1, −1), ( , . . . , , − ) 2 2 2 2 2 if m = 2M . Remark that we also consider (1, . . . , 1), (m odd) and (1, . . . , 1, 0), (1, . . . , ±1), (m even) to be fundamental. Strictly speaking they are not fundamental in the standard sense because they can be realized inside tensor products of the other (standard) fundamental weights. All other irreducible representations of Spin(m) can be built from these fundamental representations by a procedure called Cartan product. Let (V, R) and (V ′ , R′ ) be irreducible representations with weights l = (l1 , . . . , lM ) and ′ l′ = (l1′ , . . . , lM ) and corresponding weight vectors wl and wl′ . Then the tensor product (V ⊗ V ′ , R ⊗ R′ ) is also a representation of Spin(m) and usually splits in a lot of irreducible subpieces. However one piece is canonically defined. By considering the action of the Weyl group on the weight decomposition of this tensor product one arrives at the highest weight oc′ curring in V ⊗ V ′ . This weight is given by (l1 + l1′ , . . . , lM + lM ) and has ′ weight vector wl ⊗wl . By a theorem of Cartan it occurs exactly once in the decomposition of V ⊗ V ′ . The projection of V ⊗ V ′ on the highest weight subspace is the Cartan product V [×]V of two irreducible representations. Now any highest weight can be written uniquely as a linear combination of our fundamental highest weights where the coefficient of the fundamental 278 Models for Irreducible Representations of Spin(m). half integer weight representation is either zero or one (in case m is even we take the convention that the weights (1, . . . , 1, −1), ( 21 , . . . , 12 , − 21 ) only occur if the last number in a general highest weight is negative). Therefore an arbitrary highest weight can be obtained in a canonical way inside a tensor product of (symmetric) tensor powers of the fundamental representations by means of the Cartan projection. If for example m = 2M + 1, the irreducible representation s1 (1, 0, . . . , 0) + s2 (1, 1, 0, . . . , o) + · · · + sM (1, . . . , 1) can be realized inside Es1 ,...,sM = Syms1 (1, 0, . . . , 0) ⊗ Syms2 (1, 1, 0, . . . , 0) ⊗ · · · ⊗ SymsM (1, . . . , 1) while the representation s1 (1, 0, . . . , 0)+s2 (1, 1, 0, . . . , 0)+· · ·+sM (1, . . . , 1) +( 21 , . . . , 12 ) can be realized inside 1 1 Es′ 1 ,...,sM = Es1 ,...,sM ⊗ ( , . . . , ), 2 2 or in the submodule 1 1 (l1 , . . . , lM ) ⊗ ( , . . . , ). 2 2 Let E be a representation space of Spin(m) corresponding to a representation R. The Lie algebra of Spin(m) can be identified with the space Rm,2 of bivectors in Rm . Its infinitesimal representation is given by 1 dR(w)f = lim (R(exp(ǫw) − 1)f. ǫ→0 ǫ The Casimir operator of the representation R is then defined by C(R) = 1X dR(eij )2 . 4 i<j The Casimir operator C(R) acts by scalar multiplication on the R-irreducible pieces occurring in E. Its spectrum depends only on the highest weights characterizing the irreducible pieces and not on the specific way how the irreducible pieces are realized inside E. But there is no 1 − 1 correspondence between eigenspaces of the Casimir operator and highest weights because different highest weights can produce the same eigenvalue for the action of the Casimir operator. This has to do with the fact that also higher order operators (which commute with all Spin(m)-invariant operators) are needed to determine the highest weights in a unique way. However, on the canonical representation space E = Es1 ,...,sM the Casimir operator C(R) behaves much better. We know that Es1 ,...,sM = (s1 + · · · + sM , s2 + · · · + sM , . . . , sM ) ⊕ lower highest weights . P. Van Lancker, F. Sommen, D. Constales 279 Now C(R) acts by scalar multiplication on each irreducible submodule occurring in Es1 ,...,sM . In particular C(R) acts by multiplication with some constant Cs1 ,...,sM on the leading weight space. It can be proved (see [FH]) that the action of C(R) on the remaining highest weights is scalar multiplication with constants which are all different from Cs1 ,...,sM . This means that inside Es1 ,...,sM the irreducible representation (s1 + · · · + sM , . . . , sM ) is completely determined by the action of the Casimir operator. The same is also true for the realization of (s1 + · · · + sM + 21 , . . . , sM + 12 ) inside Es′ 1 ,...,sM . This result will prove to be very helpfull in the sequel. We will now show how this abstract considerations can be made concrete in the language of Clifford algebra (see also [DS] and [So3]). Let s ∈ Spin(m); consider the following two unitary representations of Spin(m): H(s)P (x1 , . . . , xk ) L(s)P (x1 , . . . , xk ) = sP (s̄x1 s, . . . , s̄xk s)s̄ = sP (s̄x1 s, . . . , s̄xk s). To define representations of P in(m) one just replaces s̄ by s̃. The Hrepresentation may act on the space of harmonic polynomials and actually defines a representation of SO(m) while the L-representation will act on monogenic polynomials. Both representations can be restricted to the corresponding subspaces of homogeneous simplicial harmonic or monogenic polynomials: H(s)Pl1 ,...,lk (x1 , . . . , x1 ∧ . . . ∧ xk ) = sPl1 ,...,lk (s̄x1 s, . . . , s̄x1 ∧ . . . ∧ xk s)s̄ and L(s)Pl1 ,...,lk (x1 , . . . , x1 ∧ . . . ∧ xk ) = sPl1 ,...,lk (s̄x1 s, . . . , s̄x1 ∧ . . . ∧ xk s). In case of scalar valued simplicial harmonic polynomials the H-representation is the usual representation of SO(m). The Casimir operators corresponding to this representations were already considered in [So2]. Let Lxl ,ij = xli ∂xlj − P xlj ∂xli be the ij-momentum operator in the variable xl . Let 2 △S,xl = xl ,ij be the Laplace-Beltrami operator in the variable xl ij LP and △S,xr xs = i<j Lxr ,ij Lxs ,ij be the “mixed” Laplace-Beltrami operator. The Casimir operators of both representations are then given by 1 C(H) 4 = X (Lx1 ,ij + · · · + Lxk ,ij )2 i<j = = k X l=1 k X l=1 △S,xl + 2 X △S,xi xj X hxi , xj ih∂xi , ∂xj i− 1≤i<j≤k △S,xl + 2 1≤i<j≤k −hxj , ∂xi ihxi , ∂xj i + hxj , ∂xj i 280 while Models for Irreducible Representations of Spin(m). 1 C(L) 4 = = 1 (Lx1 ,ij + · · · + Lxk ,ij + eij )2 2 i<j X k X m(m − 1) 1 Γxi − C(H) + . 4 8 i=1 From this it easily follows that the spaces of harmonic and monogenic simplicial polynomials are eigenspaces of the C(H) and C(L) Casimir operators respectively. The action of 41 C(H) on HlN1 k,...,lk [x1 , . . . , xk ] produces the eigenvalue k X lj (lj + m − 2j) , − j=1 k while the action of 41 C(L) on MN l1 ,...,lk [x1 , . . . , xk ] gives the eigenvalue − k X lj (lj + m − 2j + 1) − j=1 m(m − 1) . 8 Models for fundamental representations can be realized inside the complex Clifford algebra Cm . To see this, consider the actions of Spin(m) on Cm given by l(s)a = sa or h(s)a = sas̄. This leads to the fundamental representations l of Spin(m) on spinor spaces S and h of Spin(m) on k-vector spaces Cm,k , k ≤ M . In the odd dimensional case we consider the basic isotropic vectors Tj = 1 1 (e2j−1 − ie2j ), T̄j = − (e2j−1 + ie2j ) 2 2 and the idempotents Ij = Tj T̄j ; then the product I = I1 . . . IM is primitive idempotent; the ideal C+ m I is minimal and gives a model for the spinor space (see e.g. [DSS]). The action of the maximal torus gives l(s)I = = 1 exp (t1 e12 + · · · + tM e2M −1,2M )I 2 i exp (t1 + · · · + tM )I; 2 and the weight is given by ( 21 , . . . , 21 ). Next for the representation h one uses the null-k-vectors T1 ∧ · · · ∧ Tk , k = 1, . . . , M and the representation is given by h(s)T1 ∧ · · · ∧ Tk = sT1 ∧ · · · ∧ Tk s̄ = exp i(t1 + · · · + tk )T1 ∧ · · · ∧ Tk P. Van Lancker, F. Sommen, D. Constales 281 so that the weights are given by (1, 0, . . . , 0), (1, 1, 0, . . . , 0), . . . , (1, . . . , 1). Alternatively, these fundamental representations can also be realized by the highest weight vectors hx1 , T1 i, hx1 ∧x2 , T1 ∧T2 i, . . . , hx1 ∧· · ·∧xm , T1 ∧· · ·∧ TM i. These weight vectors then simply generate the spaces of 1- up to M linear alternating forms. To produce highest weight vectors for irreducible representations where the numbers determining the weight are all equal we now take symmetric tensor powers of these fundamental highest weight vectors T1 , . . . , T1 ∧· · ·∧Tm or equivalently of hx1 , T1 i, . . . , hx1 ∧· · ·∧xM , T1 ∧ · · · ∧ TM i. This may be done in a concrete way using polynomial functions of simplicial variables hx1 ∧ · · · ∧ xk , T1 ∧ · · · ∧ Tk isk on which the spin group acts like H(s)F (x1 ∧ · · · ∧ xk ) = F (s̄x1 ∧ · · · ∧ xk s) and the weight is found from the action of the maximal torus on this highest weight vector H(s)hx1 ∧ · · · ∧ xk , T1 ∧ · · · ∧ Tk isk = exp(si(t1 + · · · + tk )) hx1 ∧ · · · ∧ xk , T1 ∧ · · · ∧ Tk is , i.e. the weight is given by (s, s, . . . , s, 0, . . . , 0) (where s appears k times). It is not hard to see that this highest weight vector is simplicially harmonic or equivalently, harmonic of a matrix variable. The space of simNk plicially harmonic functions Hs,...,s,0,...,0 [x1 , . . . , xk ] is then the irreducible space to which this highest weight vector belongs. Models for all irreducible representations with integer weight are obtained by taking further tensor products of these highest weight vectors, i.e. by considering the simplicial functions F (x1 , · · · , x1 ∧ · · · ∧ xM ) = hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM , whereby the representation of Spin(m) on simplicial scalar functions is given by H(s)F (x1 , · · · , x1 ∧ · · · ∧ xM ) = F (s̄x1 s, . . . , s̄x1 ∧ · · · ∧ xM s). For the action of the maximal torus on the highest weight vectors we find H(s)hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM = exp i((s1 + · · · + sM )t1 + · · · + sM tM )hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM 282 Models for Irreducible Representations of Spin(m). i.e. the weight is given by (s1 + · · · + sM , s2 + · · · + sM , . . . , sM ). Models for all irreducible representations with half integer weight are now easily obtained by multiplying this highest weight vector further with the primitive idempotent I , i.e. to consider the spinor valued function F (x1 , · · · , x1 ∧ · · · ∧ xM ) = hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM I whereby the representation of Spin(m) on spinor valued simplicial functions is given by L(s)F (x1 , · · · , x1 ∧ · · · ∧ xM ) = sF (s̄x1 s, . . . , s̄x1 ∧ · · · ∧ xM s). For the action of the maximal torus on the highest weight vector we obtain L(s)hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM I 1 1 = exp i((s1 + · · · + sM + )t1 + · · · + (sM + )tM ) 2 2 hx1 , T1 is1 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM I i.e. the weight is (s1 + · · · + sM + 12 , . . . , sM + 21 ). If we make this choice for the highest weight vectors, there are the following observations that can be made in the odd dimensional case. Using the H-representation each weight vector wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , TM ) = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 il2 −l3 . . . hx1 ∧ . . . ∧ xM T1 ∧ . . . ∧ TM ilM and the corresponding irreducible representation belong to one space HlN1 M ,...,lM [x1 , . . . , xM ] of simplicial harmonic polynomials. This however is not enough to conclude that this space of simplicial harmonic polynomials itself is irreducible. To establish this we need to go back to our construction of the irreducible representation (l1 , . . . , lM ). By Cartan projection this irreducible representation is canonically realized inside a tensor product of symmetric powers of the fundamental representations. This tensor product contains in particular the space of simplicial harmonic polynomials HlN1 M ,...,lM [x1 , . . . , xM ]. Let Cl1 ,...,lM be the eigenvalue of the Casimir operator C(H) acting on the irreducible representation (l1 , . . . , lM ). As pointed out before, inside this tensor product Ker(C(H) − Cl1 ,...,lM ) ∼ = (l1 , . . . , lM ). Because the highest weight vectors belong to exactly one space of harmonic polynomials of simplicial type and these polynomials are already eigenspaces of C(H) we thus obtain ∼ HlN1 M ,...,lM [x1 , . . . , xM ] = (l1 , . . . , lM ) where we consider scalar valued polynomials. If we now consider the Lrepresentation, then wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , TM )I1 . . . IM = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 il2 −l3 . . . hx1 ∧ . . . ∧ xM T1 ∧ . . . ∧ TM ilM I1 . . . IM , P. Van Lancker, F. Sommen, D. Constales 283 together with the irreducible representation it generates under the action M of L, belongs to exactly one space MN l1 ,...,lM [x1 , . . . , xM ] of simplicial monogenic polynomials. Because this space is an eigenspace of the Casimir operator C(L), it is now sufficient to consider spinor valued simplicial monogenic polynomials and to apply the same line of thinking as for the Hrepresentation. We thus obtain 1 1 M ∼ MN l1 ,...,lM [x1 , . . . , xM ] = (l1 + , . . . , lM + ) 2 2 where the polynomials under consideration take values in the spinors. In the even dimensional case (m = 2M ) the construction of highest weight vectors is similar except for the fact that there are now two inequivalent spinor spaces which lead to inequivalent basic representations of Spin(m) namely + the spinor spaces C+ m I+ and Cm I− whereby the primitive idempotents I+ and I− are given by ′ I+ = I1 . . . IM −1 IM , I− = I1 . . . IM −1 IM , and 1 (1 + iem−1 em ). 2 This has to do with the fact that the pseudoscalar E = e1 . . . em is actually Spin(m)-invariant and has square (−1)M . Hence there are two invariant projectors ′ IM = T̄M TM = P+ = 1 1 (1 + (−i)M E) and P− = (1 − (−i)M E) 2 2 onto the eigenspaces of E and, as we also have that I+ = P+ I+ and I− = + P− I− , the spinor spaces C+ m I+ and Cm I− are inequivalent under the action of the representation l of Spin(m). The weights are obtained from the action of the maximal torus and given by ( 12 , · · · , 21 ) resp. ( 21 , · · · , − 12 ). Remark ∼ + ∼ + that we also have that C− m I+ = Cm em I+ = Cm I− em as equivalent spin representations. In the same way, the space of M -vectors in Cm splits into two inequivalent representations. The M -null frame T1 ∧ · · · ∧ TM satisfies P+ (T1 ∧ · · · ∧ TM ) = T1 ∧ · · · ∧ TM and P− (T1 ∧ · · · ∧ TM ) = 0 while the M -null frame T1 ∧ · · · ∧ T̄M satisfies P− (T1 ∧ · · · ∧ T̄M ) = T1 ∧ · · · ∧ T̄M and P+ (T1 ∧ · · · ∧ T̄M ) = 0. These null frames provide weight vectors for representations of weight (1, . . . , 1, 1) and (1, . . . , 1, −1) respectively. In terms of M -linear alternating forms F (x1 ∧ · · · ∧ xM ), these representations are given by forms satisfying respectively the scalar system of equations P− (∂x1 ∧ · · · ∧ ∂xM )F (x1 ∧ · · · ∧ xM ) = 0 284 Models for Irreducible Representations of Spin(m). and P+ (∂x1 ∧ · · · ∧ ∂xM )F (x1 ∧ · · · ∧ xM ) = 0, generating together the space of M -linear alternating forms. For the construction of models for irreducible representations of Spin(m) with half integer weight we now use two types of highest weight vectors F+ F− = hx1 , T1 is1 hx1 ∧ x2 , T1 ∧ T2 is2 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ TM isM I+ , = hx1 , T1 is1 hx1 ∧ x2 , T1 ∧ T2 is2 · · · hx1 ∧ · · · ∧ xM , T1 ∧ · · · ∧ T̄M isM I− , and the corresponding weights are (s1 + · · · + sM + 12 , · · · , ±(sM + 12 )). To obtain models for irreducible representations with integer weights one just leaves away the factors I+ , I− in the above definition of F+ , F− . For this choice of the highest weight vectors we now have that in case of the H-representation both the weight vectors wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , TM ) = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 il2 −l3 . . . hx1 ∧ · · · ∧ xM T1 ∧ · · · ∧ TM ilM and wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , T̄M ) = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 il2 −l3 . . . hx1 ∧ · · · ∧ xM T1 ∧ · · · ∧ T̄M ilM belong to one space HlN1 M ,...,lM [x1 , . . . , xM ] of simplicial harmonic polynomials. Now the representation s1 (1, 0, . . . , 0) + s2 (1, 1, 0, . . . , 0) + · · · + sM (1, . . . , 1, ±1) is realized inside Es1 ,...,sM −1 ,±sM = Syms1 (1, 0, . . . , 0) ⊗ Syms2 (1, 1, 0, . . . , 0) ⊗ · · · ⊗ SymsM (1, . . . , 1, ±1). In case sM = 0, the last symmetric tensor power in the above tensor product does not occur and we can immediately apply the argument with the Casimir operator as in the odd dimensional case, i.e. ∼ HlN1 M ,...,lM −1 ,0 [x1 , . . . , xM ] = (l1 , . . . , lM −1 , 0). Consider now the case where the last number in the weight is positive. Because (1, . . . , 1, +1) generates only half of the M -linear alternating forms, the space SymsM (1, . . . , 1, +1) can not be identified with the space of simplicial polynomials PsNMM,...,sM [x1 , . . . , xM ]. This means that we cannot embed simplicial harmonics directly into Es1 ,...,sM −1 ,sM . The extra conditions P. Van Lancker, F. Sommen, D. Constales 285 are found as follows. By the Capelli identity the generalized Euler operator hx1 ∧ · · · ∧ xM , ∂x1 ∧ · · · ∧ ∂xM i can be expressed as   · · · hx1 , ∂xM i hx1 , ∂x2 i hx1 , ∂x1 i + M − 1  hx2 , ∂x2 i + M − 2 · · · hx2 , ∂xM i  hx2 , ∂x1 i   det   .. .. ..   . . . hxM , ∂x2 i hxM , ∂x1 i ··· hxM , ∂xM i where the determinant of the M × M -matrix of non commuting variables Xij is given by X signσXσ(1)1 · · · Xσ(M )M . det (Xij ) = σ∈SM Because simplicial polynomials P (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xM ) are annihilated by the vector fields hxi , ∂xj i, j > i, it follows that only the diagonal elements of this matrix contribute: hx1 ∧ · · · ∧ xM , ∂x1 ∧ · · · ∧ ∂xM iPl1 ,...,lM (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xM ) = M Y (lj + M − j)Pl1 ,...,lM (x1 , x1 ∧ x2 , . . . , x1 ∧ · · · ∧ xM ), j=1 lj being the degree of homogeneity in xj . For ∂x1 ∧ · · · ∧ ∂xM acting on simplicial polynomials we have the Fischer decomposition M M PlN1 ,...,l [x1 , . . . , xM ] = (PlN1 ,...,l [x1 , . . . , xM ] ∩ Ker (∂x1 ∧ · · · ∧ ∂xM )) M M M ⊕⊥ x1 ∧ · · · ∧ xM PlN1 −1,...,l [x1 , . . . , xM ]. M −1 Let now P be a polynomial which belongs to the first space in the above decomposition. As the weight vector wl1 ,...,lM reproduces the space of simM plicial polynomials PlN1 ,...,l [x1 , . . . , xM ], it follows that the Fischer inner M product h∂x1 ∧ · · · ∧ ∂xM wl1 ,...,lM (x1 , . . . , xM ; ·), P (·)i = 0. Thus by the above Fischer decomposition ∂x1 ∧ · · · ∧ ∂xM wl1 ,...,lM (x1 , . . . , xM ; u1 , . . . , uM ) = Qu1 ∧ · · · ∧ uM for some simplicial polynomial Q. This polynomial can be identified using the identity for the action of the generalized Euler operator on simplicial polynomials, i.e. ∂x1 ∧ · · · ∧ ∂xM wl1 ,...,lM (x1 , . . . , xM ; u1 , . . . , uM ) M Y (lj + M − j)wl1 −1,...,lM −1 (x1 , . . . , xM ; u1 , . . . , uM ) = j=1 u1 ∧ · · · ∧ uM . 286 Models for Irreducible Representations of Spin(m). Remark that in this way we can also inbed the integer weight representations of Spin(m) in spaces of Cm,M -valued simplicial monogenic polynomials. The identity above now clearly shows which conditions must be imposed on the simplicial harmonic polynomials to embed them in the tensor product Es1 ,...,sM −1 ,+sM . As a matter of fact, these polynomials must be null solutions of the scalar system of equations determined by the components of P− (∂x1 ∧ · · · ∧ ∂xM ). Now we are in a situation to follow the same line of thinking as in the odd dimensional case and we have ∼ HlN1 M ,...,lM [x1 , . . . , xM ] ∩ Ker P− (∂x1 ∧ · · · ∧ ∂xM ) = (l1 , . . . , lM −1 , +lM ) ∼ HlN1 M ,...,lM [x1 , . . . , xM ] ∩ Ker P+ (∂x1 ∧ · · · ∧ ∂xM ) = (l1 , . . . , lM −1 , −lM ), and ∼ HlN1 M ,...,lM [x1 , . . . , xM ] = (l1 , . . . , lM −1 , +lM )⊕ (l1 , . . . , lM −1 , −lM ), (lM > 0). Of course this characterization remains true if lM = 0, but now the extra systems of scalar equations are satisfied in a trivial way and are actually redundant. In case lM > 0, this splitting is very natural because HlN1 M ,...,lM [x1 , . . . , xM ] is actually an irreducible P in(m)-module corresponding to the weight (l1 , . . . , lM ). The important fact here is that in case of the P in-representation either the M -frame T1 ∧ · · · ∧ TM or T1 ∧ · · · ∧ T̄M generate the whole space of M -linear alternating forms, so the conditions arising from the components of P− (∂x1 ∧ · · · ∧ ∂xM ) or P+ (∂x1 ∧ · · · ∧ ∂xM ) do not occur. By regarding the irreducible P in(m)-module HlN1 M ,...,lM [x1 , . . . , xM ] (lM > 0) as a Spin(m)-module, it splits as a sum of two irreducible Spin-modules: (l1 , . . . , lM −1 , +lM ) ⊕ (l1 , . . . , lM −1 , −lM ). The two extra Spin(2M )-invariant systems of scalar equations then simply identify the sign of the last number in the weight. Let us consider as an example the one variable case: m = 2, M = 1. Vectors x ∈ R2 are written as x = e1 x1 + e2 x2 while the Dirac operator is given by ∂x = e1 ∂x1 + e2 ∂x2 . Irreducible representations of P in(2) are labelled by numbers k ∈ N and the corresponding models are given by harmonic polynomials (in R2 ) which are homogeneous of degree k. As Spin(2)-representations, the spaces of homogenous harmonic polynomials split into two pieces given by the kernel of the components of P− ∂x or P+ ∂x . Now 1 − ie12 )∂x 2 1 + ie12 )∂x P− ∂x = ( 2 P+ ∂x = ( = = e1 − ie2 (∂x1 + i∂x2 ) 2 e1 − ie2 (∂x1 − i∂x2 ) 2 12 (∂x21 +∂x22 ). Hence, as irreducible Spin(2)-modules, and P+ ∂x P− ∂x = −1+ie 2 the harmonic polynomials should be annihilated by the appropiate Cauchy- P. Van Lancker, F. Sommen, D. Constales 287 Riemann operators: ∼ (+k) = (−k) ∼ = anti holomorphic polynomials homogeneous of degree k holomorphic polynomials homogeneous of degree k. In case of the L-representation we can follow the same procedure. Both the weight vectors wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , TM )I1 . . . IM = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 l2 −l3 i . . . hx1 ∧ · · · ∧ xM T1 ∧ · · · ∧ TM ilM I1 . . . IM and ′ wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , T̄M )I1 . . . IM = hx1 T1 il1 −l2 hx1 ∧ x2 T1 ∧ T2 il2 −l3 . . . ′ hx1 ∧ · · · ∧ xM T1 ∧ · · · ∧ T̄M ilM I1 . . . IM M belong to the same space PlN1 ,...,l [x1 , . . . , xM ] of simplicial monogenic polyM nomials. Now the representation s1 (1, 0, . . . , 0) + s2 (1, 1, 0, . . . , 0) + · · · + sM (1, . . . , 1, ±1) +( 21 , . . . , 12 , ± 21 ) = (l1 + 21 , . . . , lM −1 + 21 , ±(lM + 12 )) can be realized inside 1 1 1 Es′ 1 ,...,sM −1 ,±sM = Es1 ,...,sM −1 ,±sM ⊗ ( , . . . , , ± ), 2 2 2 or in the submodules of spinor valued simplicial harmonics 1 1 1 (l1 , . . . , lM −1 , +lM ) ⊗ ( , . . . , , + ) and 2 2 2 1 1 1 (l1 , . . . , lM −1 , −lM ) ⊗ ( , . . . , , − ) 2 2 2 As in the integer weight case, we now immediately have for lM = 0: 1 1 1 M ∼ PlN1 ,...,l [x1 , . . . , xM ] (C+ m I+ − valued) = (l1 + , . . . , lM −1 + , + ) M −1 ,0 2 2 2 1 1 1 M ∼ PlN1 ,...,l [x1 , . . . , xM ] (C+ m I− − valued) = (l1 + , . . . , lM −1 + , − ), M −1 ,0 2 2 2 i.e., the ± 12 at the end of the weights are distinguished by considering the appropiate spinor values. Let us now consider the case where the last number in the weight is positive. To embed the spinor valued (corresponding to I) simplicial monogenics in El′1 ,...,lM −1 ,+lM we have to impose the condition 288 Models for Irreducible Representations of Spin(m). that its harmonic components belong to the right space of simplicial harmonics. This means that these simplicial monogenics should be annihilated by the Spin(2M )-invariant scalar system of equations determined by the components of P− (∂x1 ∧ · · · ∧ ∂xM ). Applying now the argument with the Casimir operator we thus get for spinor valued (corresponding to C+ m I+ ) polynomials: M PlN1 ,...,l [x1 , . . . , xM ] ∩ Ker ( components of P− (∂x1 ∧ · · · ∧ ∂xM )) M 1 1 1 ∼ = (l1 + , . . . , lM −1 + , +(lM + )). 2 2 2 or in case of spinor values corresponding to C+ m I− : M PlN1 ,...,l [x1 , . . . , xM ] ∩ Ker ( components of P+ (∂x1 ∧ · · · ∧ ∂xM )) M 1 1 1 ∼ = (l1 + , . . . , lM −1 + , −(lM + )). 2 2 2 We now conjecture that in both cases the system of scalar equations determined by the components of P− (∂x1 ∧ · · · ∧ ∂xM ) and P+ (∂x1 ∧ · · · ∧ ∂xM ) are actually redundant; they should be satisfied automatically by simplicial monogenics taking values in the appropiate spinor spaces. Now M PlN1 ,...,l [x1 , . . . , xM ] (Cm I+ valued) is an irreducible P in(m)-module corM responding to the weight (l1 + 12 , . . . , lM −1 + 21 , lM + 12 ). Regarding it as a Spin(m)-module, it splits as a direct sum of the two irreducible Spinrepresentations (l1 + 21 , . . . , lM −1 + 12 , lM + 21 ) and (l1 + 12 , . . . , lM −1 + 1 1 2 , −(lM + 2 )), lM > 0. On the level of Clifford algebra, this splitting comes from the decomposition of the values of the space of Cm I+ -valued simpli+ − ∼ + cial monogenics: Cm I+ = C+ m I+ ⊕ Cm I+ = Cm I+ ⊕ Cm I− em . Clearly the + space of Cm I+ -valued simplicial monogenics contains the weight vector wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , TM )I1 . . . IM while the space of C+ m I− em -valued simplicial monogenics contains the inequivalent weight vector ′ em . wl1 ,...,lM (x1 , . . . , xm ; T1 , . . . , T̄M )I1 . . . IM Hence for simplicial monogenics or half integer weight representations the consideration of the appropiate values makes the “extra” scalar system of equations needed in the integer weight case superfluous. We thus conclude: M PlN1 ,...,l [x1 , . . . , xM ] ∼ = (l1 + 21 , . . . , lM −1 + 12 , +(lM + 21 )) M where the polynomials are C+ m I+ -valued, and M PlN1 ,...,l [x1 , . . . , xM ] ∼ = (l1 + 21 , . . . , lM −1 + 12 , −(lM + 21 )) M where the polynomials are C+ m I− -valued. P. Van Lancker, F. Sommen, D. Constales 289 References [ABLSS] W. W. Adams, C. A. Berenstein, P. Loustaunau, I. Sabadini, D. C. Struppa: Regular functions of several quaternionic variables, to appear in J. Geom. Anal. [BDS] F. Brackx, R. Delanghe and F. Sommen: Clifford Analysis, Pitman, London, 1982. [Co] D. Constales: The relative position of L2 -domains in complex and Clifford analysis, Ph. D. thesis, State Univ. Ghent, 1989-1990. [RSS] R. Delanghe, F. Sommen, V. Souček: Clifford analysis and spinor valued functions, Kluwer Acad. Publ., Dordrecht, 1992. [DS] R. Delanghe, V. Souček: On the structure of spinor valued differential forms, Complex Variables: Theory Appl. 20, 1992, pp. 171-184. [FH] W. Fulton, J. 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