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The Howe dual pair in Hermitean Clifford analysis. (English) Zbl 1201.30061
In this paper, the classical Clifford analysis is additionally furnished with a complex structure. The Riemannian setting is now replaced by Kähler geometry. Unfortunately, the rotation invariance of the null solutions of the Dirac operator is not longer valid. The authors study a new associated Dirac operator $$\partial_J$$. With Hermitian analysis, simultaneously null solutions of both operators $$\partial$$ and $$\partial_J$$ are considered. In this way, the rotation invaraince reduces to $$U(n)$$-invariance. The main interest of the paper lies in the consideration of the Howe dual pair with respect to Hermitean Clifford analysis. Spinor valued polynomials play an important role. Fischer decompositions for Hermitian monogenic functions are re-classified.

MSC:
 30G35 Functions of hypercomplex variables and generalized variables 15A66 Clifford algebras, spinors
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References:
 [1] Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F. and Souček, V.: Fundaments of Hermitean Clifford analysis. I. Complex structure. Compl. Anal. Oper. Theory 1 (2007), no. 3, 341-365. · Zbl 1131.30019 [2] Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F. and Souček, V.: Fundaments of Hermitean Clifford analysis. II. Splitting of $$h$$-monogenic equations. Complex Var. Elliptic Equ. 52 (2007), no. 10-11, 1063-1079. · Zbl 1144.30019 [3] Brackx, F., Delanghe, R. and Sommen, F.: Clifford analysis. Research Notes in Mathematics 76 . Pitman, Boston, MA, 1982. · Zbl 0529.30001 [4] Brackx, F., De Knock, B., De Schepper, H., and Sommen, F.: On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis. Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 395-416. · Zbl 1182.30081 [5] Brackx, F., De Knock, B., De Schepper, H.: A matrix Hilbert transform in Hermitean Clifford analysis. J. Math. Anal. Appl. 344 (2008), no. 2, 1068-1078. · Zbl 1148.44004 [6] Brackx, F., De Schepper, H., De Schepper, N. and Sommen, F.: Hermitean Clifford-Hermite polynomials. Adv. Appl. Clifford Algebr. 17 (2007), no. 3, 311-330. · Zbl 1134.30039 [7] Brackx, F., De Schepper, H. and Sommen, F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Algebr. 18 (2008), no. 3-4, 451-487. · Zbl 1177.30064 [8] Brackx, F., De Schepper, H. and Sommen, F.: A theoretical framework for wavelet analysis in a Hermitean Clifford setting. Commun. Pure Appl. Anal. 6 (2007), no. 3, 549-567. · Zbl 1149.30036 [9] Colombo, F., Sabadini, I., Sommen, F. and Struppa, D.C.: Analysis of Dirac systems and computational algebra. Progress in Mathematical Physics 39 . Birkhäuser Boston, Boston, MA, 2004. · Zbl 1064.30049 [10] Delanghe, R., Sommen, F. and Souček, V.: Clifford algebra and spinor-valued functions, a function theory for the Dirac operator. Kluwer Academic Publishers, Dordrecht, 1992. · Zbl 0747.53001 [11] Eelbode, D.: Stirling numbers and spin-Euler polynomials. Experiment. Math. 16 (2007), no. 1, 55-66. · Zbl 1201.30064 [12] Eelbode, D.: Irreducible $$\mathfraksl(m)$$-modules of Hermitean monogenics. Complex Var. Elliptic Equ. 53 (2008), no. 10, 975-987. · Zbl 1159.30029 [13] Fulton, W. and Harris, J.: Representation theory. A first course , 3rd edition. Springer Verlag, New York, 1996. · Zbl 0744.22001 [14] Gilbert, J. and Murray, M.: Clifford algebra and Dirac operators in Harmonic analysis. Cambridge Studies in Advanced Mathematics 26 . Cambridge University Press, Cambridge, 1991. · Zbl 0733.43001 [15] Goodman, R. and Wallach, N.R.: Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications 68 . Cambridge University Press, Cambridge, 1998. · Zbl 0901.22001 [16] Goodman, R.: Multiplicity-free spaces and the Schur-Weyl-Howe duality. In Representations of real and p-adic Groups , 305-415. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 2 . World Scientific, Singapore, 2004. · Zbl 1060.22010 [17] Gürlebeck, K. and Sprössig, W.: Quaternionic and Clifford calculus for physicists and engineers. J. Wiley & Sons, Chichester, 1997. · Zbl 0897.30023 [18] Howe, R.: Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535-552. JSTOR: · Zbl 0716.22006 [19] Howe, R.: Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313 (1989), no. 2, 539-570. · Zbl 0674.15021 [20] Howe, R.: Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons. In Applications of groups theory in physics and mahtematical physics (Chicago, 1982) , 179-207. Lectures in Appl. Math. 21 . Amer. Math. Soc., Providence, RI, 1985. · Zbl 0558.22018 [21] Rocha-Chavez, R., Shapiro, M. and Sommen, F.: Integral theorems for functions and differential forms in $$\mathbbC_m$$. Chapman & Hall/CRC Research Notes in Mathematics 428 . Chapman&Hall / CRC, Boca Raton, FL, 2002. · Zbl 0991.32002 [22] Sabadini, I. and Sommen, F.: Hermitian Clifford analysis and resolutions. Math. Methods Appl. Sci. 25 (2002), no. 16-18, 1395-1413. · Zbl 1013.30033 [23] Sommen, F. and Peña Peña, D.: A Martinelli-Bochner formula for the Hermitian Dirac equation. Math. Methods Appl. Sci. 30 (2007), no. 9, 1049-1055. · Zbl 1117.30040
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