Nolan R. Wallach

Curriculum Vitae

Personal

Address:                                 University of California, San Diego

Department of Mathematics

La Jolla, California  92093-0112

(858) 534-2645

Degrees                                  B.S.            University of Maryland   1962

M.A.          Washington University    1963

Ph.D.         Washington University    1966

Professional

1989-                                       Professor of Mathematics, Above Scale

University of California, San Diego

1986-1989                                Hermann Weyl Professor of Mathematics

1972-1986                                Professor of Mathematics

1970-1972                                Associate Professor of Mathematics

1969-1970                                Assistant Professor of Mathematics

Rutgers University

1968-1969                                Lecturer of Mathematics

1966-1968                                Instructor of Mathematics

University of California, Berkeley

Hour Address, Annual meeting of the AMS, January, 1975.

Invited Address, International Congress of Mathematicians, Helsinki, August, 1978.

Hour Address, Joint MAA-AMS session, Annual meeting of the AMS, January, 1990.

Plenary Address, Japan Mathematical Society, Annual meeting, September, 1990.

Major Professional Activity

Member, National Mathematics Committee, 1985-1992

Member, Editorial Boards Committee of the AMS (Chair 1992), 1991-1993

NSF-Postdoctoral Selection Committee (Chair 1994), 1992-1994

NYI-Selection Committee, 1993

Editor, RES Section, Bulletin of the AMS, 1995-1996

Associate Editor, Bulletin of the AMS, 1996-1998

Member of the committee: U.S. Mathematical Sciences Research Institutes:

Value and Need, The Board of Mathematical Sciences of the NAS 1998.

Associate Editor, Annals of Mathematics, 1997-2003

Council of the AMS, 1998-2001

Science Policy committee of the AMS, 1998-2001

Editor Mathematics: Theory and Applications, Birkhauser, Boston, 2001-

Science Policy committee of the AMS, 2007-

Prizes and Fellowships

Alfred Sloan Fellowship 1972-1974

Linback Award for Research Excellence, 1977

Honorary Professor, University of Cordoba, Argentina, 1989

Elected Fellow of the American Academy of Arts and Sciences, 2004

Publications of Nolan R. Wallach

1. N. R. Wallach:  “On maximal subsystems of root systems,” Canad. J. Math., 20 (1968), 555-574.

2. N. R. Wallach:  “Induced representation of Lie algebras and a theorem of Borel Weil,” Trans. AMS, 136 (1969), 181-187.

3. N. R. Wallach:  “Induced representations of Lie algebras II,” Proc. AMS, 21 (1969), 161-166.

4. N. R. Wallach and M. do Carmo:  “Minimal immersions of  spheres into spheres,” Proc. Natl. Academy of Sci., 63 (3) (1969), 640-642.

5. N. R. Wallach and M. Do Carmo:  “Minimal immersions of  sphere bundles over spheres,” Proc. of Brazilian Acad. Sci., 42 (1) (1970), 5-9.

6. N. R. Wallach:  “Extension of locally defined minimal immersions into spheres,” Archiv der Math., XXI (1970), 209-213.

7. N. R. Wallach and M. Cahen:  Lorentzian symmetric spaces,” Bull. AMS, 76 (3) (1970), 585-591.

8. N. R. Wallach:  “Homogeneous positively pinched Riemannian manifolds,” Bull. AMS, 76 (1970), 783-786.

9. N. R. Wallach and M. do Carmo:  “Representations of  compact groups and minimal immersions into spheres,” J. Diff. Geom., 4 (1970), 91-104.

10. N. R. Wallach and F. Warner:  “Curvature forms on 2-manifolds,” Proc. AMS, 25 (3) (1970), 712-713.

11. N. R. Wallach:  “Cyclic vectors and irreducibility for principal series representations,” Trans. AMS, 158 (1971), 107-113.

12. N. R. Wallach:  “Application of the higher osculating spaces to the spherical principal series,” J. Diff. Geom., 5 (1971), 405-413.

13. N. R. Wallach and M. do Carmo:  “Minimal immersions of  spheres into spheres,” Annals of Math., 93 (1971), 43-62.

14. N. R. Wallach and S. Greenfield:  Extendability properties of submanifolds of C2,” Proc. of 1970 Carolina Conf. on Holomorphic Mappings and Minimal Surfaces, (1971), 77-85.

15.N. R. Wallach and S. Greenfield:  “The Hilbert ball and bi-ball are holomorphically inequivalent,” Bull. AMS, 77 (2) (1971), 261-263.

16. N. R. Wallach:  “Minimal Immersions of Symmetric spaces,” Marcel Dekker Series of  Pure and Applied Math., 8 (1972), 1-40.

17. N. R. Wallach and S. Greenfield:  “Global hypoellipticity and Liouville numbers,” Proc. AMS, 31 (1) (1972), 112-114.

18. N. R. Wallach and S. Greenfield:  Automorphism groups of bounded domains in Banach spaces,” Trans. AMS,  166 (1972), 45-57.

19. N. R. Wallach:  “Three new examples of compact manifolds admitting Riemannian structures of positive curvature,” Bull. AMS, 78 (1972), 55-56.

20. N. R. Wallach:  “Cyclic vectors and irreducibility for non-unitary principal series representations,” Trans. AMS, 164 (1972), 389-396.

21. N. R. Wallach:  “Compact homogeneous Riemannian manifolds with strictly positive curvature,” Annals Math., 96 (1972), 277-295.

22. N. R. Wallach and K. Johnson:  “Composition series and intertwining operators for the spherical principal series,” Bull. AMS,  78 (1972), 1053-1059.

23. N. R. Wallach:  “Harmonic analysis on homogeneous spaces,” Marcel Dekker, New York, 1973.

24.N. R. Wallach and J. Lepowsky:  “Finite dimensional and infinite dimensional representations of linear Lie groups,” Trans. AMS, 184 (1973), 223-246.

25. N. R. Wallach and S. Greenfield:  “Remarks on global hypoellipticity,” Trans. AMS, 183 (1973), 153-164.

26.N. R. Wallach and S. Greenfield:  “Globally hypoelliptic vector fields,” Topology, 12 (1973), 247-253.

27. N. R. Wallach:  Kostants Pg and  Rg matrices and intertwining integrals,” Proc. Symp. in Pure Math., XXVI AMS, Providence, RI  (1973), 269-275.

28. N. R. Wallach:  “On Harish-Chandra's C-function,” Amer. J. Math., 97 (1975), 386-403.

In the paper below all results (especially Lemma 2.1) should exclude Sp(n,1) and rank 1 F_4.

29. N. R. Wallach and R. Hotta:  “On Matsushima's formula for the Betti numbers of a locally symmetric space,” Osaka J. Math., 12 (1975), 419-431.Get PDF version

30.N. R. Wallach and S. Aloff:  “An infinite family of 7-manifolds admitting positively curved Riemannian structures,” Bull. AMS, 81 (1975), 93-97.

31.N. R. Wallach:  “On the unitarizability of representations with highest weights,” Proceedings of Conference at Luminy of 1974 (1975).

32.N. R. Wallach:  “On an asymptotic formula of Gelfand and Gangolli for the spectrum of  G\G,” J. Diff. Geom., 11 (1976), 171-195.

33. N. R. Wallach:  “On the Selberg trace formula in the case of compact quotient,” Bull. AMS, 82 (1976), 171-195.

34.N. R. Wallach:  “On the Enright-Varadarjan modules: A construction of the discrete series,” Ann. Ec. Norm. Sup., 9 (1976), 81-101.

35.N. R. Wallach and A. Knapp:  Szego kernels associated with the discrete series,” Inventiones Math., 34 (1976), 163-200.

36.N. R. Wallach and K. Johnson:  “Composition series and intertwining operators or the spherical principal series,I,” Trans. AMS, 229 (1977), 137-173.

37.N. R. Wallach:  Symplectic Geometry and Fourier Analysis, Math. Sci. Press, 1977.

38. N. R. Wallach:  “Representations of semi-simple Lie groups and Lie algebras,” Lie Theories and Their Applications, Proceedings 1977 Canadian Math. Congress, 143-246.

39. N. R. Wallach and D. DeGeorge:  “Limit formulas for multiplicities in L2( G\G),” Ann. Math., 107 (1978), 133-150.

40. N. R. Wallach and T. J. Enright:  “The fundamental series for a semi-simple Lie algebra,” Acta Math., 140 (1978), 1-32.

41.N. R. Wallach:  “The spectrum of compact quotients of  semi-simple Lie groups,” Int. Congr. of Math., 1978.

42.N. R. Wallach:  “The analytic continuation of the discrete series, II,” Trans. AMS, 251 (1979), 19-37.

43 N. R. Wallach:  “The analytic continuation of the discrete series, I,” Trans. AMS, 251 (1979), 1-17.

44.N. R. Wallach and D. DeGeorge:  “Limit formulas for multiplicities in L2(G\G), II” Ann. Math., 109 (1979), 477-495.

45.N. R. Wallach:  “Representations of reductive Lie groups, Part I, Proc. Symp. in Pure Math., 33 (1979), 71-86.

46.N. R. Wallach and A. Borel:  “Continuous cohomology, discrete subgroups and representations of reductive groups,” Annals of Math. Studies, 94 (1980).

47.N. R. Wallach and R. Goodman:  “Conical vectors and Whittaker vectors,” J. Functional Analysis, 39 (1980), 199-279.

48. N.R.Wallach and T.J.Enright, Homological algebra and group representations, Duke Math Journal 47(1980),1-15.

49. N. R. Wallach:  “Restriction of Whittaker modules to certain parabolic subalgebras,” Proc. AMS, 81 (1981), 181-188.

50. N. R. Wallach and J. T. Stafford:  “Restriction of admissible representations to certain parabolic subalgebras,” Trans. AMS, 272 (1982), 333-350.

51. N. R. Wallach and R. Goodman:  “Classical and quantum-mechanical systems of Toda lattice type, I,” Comm. Math. Phys., 83 (1982), 355-386.

52. N. R. Wallach and A. Rocha-Caridi:  “Projective modules over graded Lie algebras, I,” Math. Zeit., 180 (1982), 151-172.

53. N. R. Wallach:  “On the constant term of an L2-automorphic form,” Operator Algebras and Group Representations.II (Momog. Stud. Math., Vol. 18), Pittman, London, 1984, 227-237.

54. N. R. Wallach, T. J. Enright, and R. Howe:  “A classification of unitary highest weight modules,” Proceedings of Utah Conference,  1982,  Burkhauser, New York (1983).

55. N. R. Wallach and A. Rocha-Caridi,  Highest weight modules over graded Lie algebras:  Resolutions, filtrations and character formulas,” Trans. AMS, 277 (1983), 133-162.

56. N. R. Wallach and A. Rocha-Caridi:  “Characters of irreducible representations of the Lie algebra of vector fields on the circle,” Invent. Math., 72 (1983), 57-76.

57. N. R. Wallach:  “Asymptotic expansions of generalized matrix entries,” Springer-Verlag Lecture Notes in Math, 1024 (1983), 287-369.

58. N. R. Wallach and J. Wolf:  “Poincare series associated with L1-discrete series,”Proceedings of Utah Conference, 1982, Birkhauser, New York (1983).

59. N. R. Wallach, T. J. Enright, V. Parthasarathy and J. Wolf: “Unitarizability of a class of derived functor modules,” Proc.Nat. Acad. Sci., 80 (1983), 7047-7050.

60. N. R. Wallach:  “L2 - Automorphic forms and cohomology classes on arithmetic quotients of SU(p.q.),” Math. Annalen, 266 (1984), 261-278.

61. N. R. Wallach and R. Goodman:  “The structure and projective unitary representations of loop groups and the group of diffeomorphisms of the circle,” J. Reine Angew. Math., 347 (1984), 69-133.

62. N. R. Wallach and A. Rocha-Caridi:  “Characters of irreducible representations of the Virasoro algebra, Math. Zeit., 185 (1984), 1-21.

63. N. R. Wallach:  “Asymptotic behavior of holomorphic representations,” Memoirs of the French Math. Soc., 2e Serie, 15 (1984), 291-305.

64. N. R. Wallach and Roe Goodman:  “Classical and quantum- mechanical systems of Toda lattice type, II,” Comm. Math., Phys., 94 (1984), 177-217. Get reprint,

65. N. R. Wallach:  “On unitarizable derived functor modules,” Inv. Math., 78 (1984), 131-141.

66. N. R. Wallach:  “Classical invariant theory and the Virasoro algebra,” in Vertex Operators in Mathematics and Physics, Springer-Verlag, 1984.

67. N. R. Wallach, T. J. Enright, V. Parthasarathy and J. Wolf: “Unitarizable derived functor modules with small spectrum,” Acta Mathematica, 154 (1985), 105-136.

68. N. R. Wallach and Roe Goodman:  “Projective unitary positive energy representations of Diff(S1),” J. Funct. Anal., 63 (1985), 299-321.

69.N. R. Wallach and Roe Goodman:  “Classical and quantum- mechanical systems of Toda lattice type, III, “ Comm. Math. Phys., 105 (1986), 473-509.

70.N. R. Wallach:  “A class of non-standard modules for affine Lie algebras,” Math. Zeit., 196 (1987), 303-313.

71.N. R. Wallach:  “On the irreducibility and inequivalence of unitary representations of gauge groups,” Comp. Math., 64 (1987), 3-29.

72. N. R. Wallach:  Real Reductive Groups, I, Academic Press Pure and Applied Mathematics, Boston, 132 (1988).

73. N. R. Wallach:  “Lie algebra cohomology and the Holomorphicic continuation of generalized Jacquet integrals,” Advanced Studies in Pure Mathematics, 14 (1988), 123-151.Get .pdf version

74. N. R. Wallach and R. Miatello:  Automorphic forms constructed from Whittaker vectors,” J. Func. Analysis, 86 (1989), 411-487.

75. N. R. Wallach and R. Goodman:  “Higher order Sugawara operators for affine Lie algebras,  Trans. AMS, 315 (1989), 1-55.

76. N. R. Wallach:  “Limit multiplicities in L2( G \G),  Lecture Notes in Mathematics 1447, Springer-Verlag (1989), 31-56.Paper in pdf form

77. N. R. Wallach and D. Vogan:  “Intertwining operators for real reductive groups,” Advances in Math., 82 (1990), 203-242.

78. N. R. Wallach and R. Miatello:  Kuznetsov formulas for real rank one groups,” J. Func. Analysis, 93 (1990), 171-205.

79. N. R. Wallach and R. Miatello:  Kuznetsov Formulas for Product Groups of R-rank One,” Israeli Mathematical Conference Proceedings, 31 (1990), 305-320.

80. N. R. Wallach:  “The powers of the resolvent on a locally symmetric space,  Bull. Soc. Math. Belg. 42 (1990), 777-795.

81. N. R. Wallach:  “Polynomial differential operators associated with a Hermitian symmetric space,  Representation Theory of Lie Groups and Lie Algebras, World Scientific, Singapore, 1992, 76-95.

82. N. R. Wallach:  Real Reductive Groups II, Academic Press, Boston, 1992.

83. N. R. Wallach with J. Tirao (co-editors):  New Developments in Lie Theory and Their Applications, Progress in Mathematics, Birkäuser, Boston, 1992.

84. N. R. Wallach:  Automorphic Forms,” in New Developments in Lie Theory and Their Applications, Birkhäuser, Boston (1992), 1-25.

85. N. R. Wallach and R. Miatello:  “The resolvent of the Laplacian on locally symmetric spaces,” J. Diff. Geom. 36 (1992), 663-698.

86. N. R. Wallach:  “Invariant differential operators on a reductive Lie algebra and Weyl group representations,” J. A.M.S  6 (1993), 779-816.

87. P. J. Sally, Jr. and N. R. Wallach (Editors):  “Representation Theory and Automorphic Forms,  Reprints from the Bulletin of AMS 2 (1993).

88. B. H. Gross and N. R. Wallach:  “A distinguished family of unitary representations for the exceptional groups of real rank 4, Lie Theory  and  Geometry, In Honor of  B. Kostant, Volume 1, Birkhäuser, Boston (1994), 289-304.

89. N. R. Wallach:  “Transfer of unitary representations between real forms,  Contemp. Math.  177 (1994), 181-216.

90. N. R. Wallach and M. Hunziker:  “On the Harish-Chandra homomorphism  of invariant differential operators on a reductive Lie algebra,”  Contemp. Math.  191 (1995), 223-243.

91. N. R. Wallach:  “On a theorem of Milnor and Thom,” Progress in Nonlinear Differential Equations,  20 (1996), 331-348.

92. J.S. Huang, T. Oshima, N. R. Wallach:  “Dimensions of spaces of generalized spherical functions,  Amer. J. Math. 118 (1996), 637-652.

93. B. Gross and N. R. Wallach:  “On quaternionic discrete series representations, and their continuations,” J. reine angew. Math., 481 (1996), 73-123

94. T. Enright and N. R. Wallach:  “Embeddings of Unitary Highest Representations and Generalized Dirac Operators,” Math. Annalen., 307 (1997), 627-646

95. R. Goodman and N. R. Wallach:  Representations and invariants of the classical groups, Cambridge University Press., Encyclopedia of Mathematics and its applications

68, Cambridge, (1998).

96. N.R. Wallach: “Cvectors”, Representations of Lie groups and quantum groups,(Trento 1993), 205-270, Confman Sci. Tech., Harlow, (1994).

97. N. R. Wallach: “A variety of solutions to the Yang-Baxter equation,” Advances in Geometry, 391-399.

98. H. Kraft, L.W. Small, N.R. Wallach: “Hereditary properties of direct summands of Algebras,” Mathematical Research Letter, 6 (1999), 371-375.

99. A. Borel and N.R. Wallach: Continuous cohomology, discrete subgroups and representations of reductive groups, 2nd. Ed., Mathematical Surveys and Monographs, 67, American Mathematical Society, RI (2000).

100. R.W. Bruggeman, R.W. Miatello, N. R. Wallach: “Resolvant and lattice points on symmetric spaces of strictly negative curvature,” Math. Ann. 315 (1999), 617-639.

101. N.R. Wallach and J. Willenbring: “On Some q-analogs of a theorem of Kostant-Rallis,” Canad. J. Math., 52 (2000), 438-448.

102. N.R. Wallach: “An Unentangled Gleason's theorem” Quantum computation and information (Washington, DC, 2000), 291--298,Contemp. Math. 305(2002). quant. - ph/002058

103. B. Gross and N. Wallach: “Restriction of small discrete series representations to symmetric subgroups,” Proceedings of Symposia in Pure Mathematics, Volume 68 (2000), 255-272.

104. H. Kraft, L.W. Small and N. Wallach: “Properties and Examples of FCR-Algebras,” Manuscripta Math. 104(2001), 443-450.

105. David A. Meyer and Nolan Wallach, Invariants for multiple qubits: the case of 3 qubits'',in Ranee K. Brylinski and Goong Chen, eds.,The Mathematics of Quantum Computation(Boca Raton: CRC Press 2002) 77-97.

106. David A. Meyer and Nolan R. Wallach, “Global entanglement in multiparticle systems”, quant-ph/0108104, Journal of Math. Phys. 43(2002),4273-4278

107. “Generalized Whittaker vectors for holomorphic and quaternionic representations”,  Commentarii Math. Helv. 79(2003), 266-307.

108.  H. Kraft and N. Wallach, “On the separation property of orbits in representation spaces. Special issue in celebration of Claudio Procesi's 60th birthday,  J. Algebra 258 (2002), 228--254

109. A.Garsia and N. Wallach, “Qsym is free over Sym”, J. Comb. Theorey Ser. A,104 (2003),  217--263.

110. A.M. Garsia and N. R. Wallach, Sime new applications of orbit harmonics, Sem. Lothar. Combin. 50(2003/2004), Art. B50j. 47pp.

111. T.Enright, M.Hunziker and N.Wallach, “A Pieri rule for Hermitian Symmetric Spaces I”, Pacific J. Math.,214(2004),23-30.

112. J.Kuttler and N. Wallach, “Representations of SL2 and the distribution of points in Pn”, Non-commutative harmonic analysis, 355-373,Prog. Math.,220, 2004 Birkhauser Boston,

113. T. Enright and N. Wallach, “A Pieri rule for Hermitian Symmetric Spaces II”, Pacific J. Math., 216(2004), 51-61..

114. Nolan R. Wallach, The Hilbert series for measures of entanglement in 4 qubits, Acta Appl. Math. 86(2005),203-220.

115. N.R. Wallach and Chen-bo Zhu, Transfer of Unitary Representations, Asian J. Math. 8(2004), 861-879.

116. A. Garsia and Nolan Wallach, “Combinatorial Aspects of the Baker Akhieser Functions for S2”, European J. Combin. 25(2004), 1231-1262.

117. J. Bell, A. Garsia and N . Wallach, Some new methods in the theory of m-quasininvariants, Electron. J. Combin. 11(2004/2006), Research Paper 20, 32 pp.

118.  Nolan R. Wallach,“Holomorphic continuation of generalized Jacquet integrals for degenerate principal series”, Represent. Theory, 10(2006), 380-396..

119. Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005),1-29.

120. Jessica Benton, Rion Snow and Nolan Wallach, A combinatorial problem associated with nanograms, Linear Algebra Appl. 412(2006) 30-38.

121. Kostant Bertram and Nolan Wallach, GelfandZeitlin theory from the perspective of classical mechanics I, Studies in Lie Theory,319-364, Progr. Math., 243, Birkhauser Boston, 2006.

121. Kostant Bertram and Nolan Wallach, GelfandZeitlin theory from the perspective of classical mechanics II, The unity of mathematics, 387-364, Progr. Math., 244, Birkhauser Boston, 2006.

122. Hanspeter Kraft and Nolan R. Wallach, On the nullcone of representations of reductive groups, Pacific J. Math. 224 (2006), 119-139.

123. A. M. Garsia and N. R. Wallach, The non-degeneracy of the bilinear form of m-quasiinvariants, Adv. In Appl. Math. 37(2006), 309-359.

124. Nolan R. Wallach, Holomorphic continuation of generalized Jacquet integrals for degenerate principal series. Represent. Theory 10 (2006), 380–398.

125. A. M.;Garsia, N. Wallach, $r$r-Qsym is free over Sym. J. Combin. Theory Ser. A 114 (2007), no. 4, 704–732.

126. Karin Baur, , Nolan Wallach, A class of gradings of simple Lie algebras. Lie algebras, vertex operator algebras and their applications, 3–15, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007.

127. Ron, Evans, Nolan Wallach,Pfaffians and strategies for integer choice games. Harmonic analysis, group representations, automorphic forms and invariant theory, 53–72, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 12, World Sci. Publ., Hackensack, NJ, 2007.

128. Gilad Gour, Nolan Wallach, Entanglement of subspaces and error correcting codes, Physical Review A, 76(2007),

129. Nolan R. Wallach,  Quantum computing and entanglement for mathematicians. Representation theory and complex analysis, 345–376, Lecture Notes in Math., 1931, Springer, Berlin, 2008.

130. Nolan R. Wallach,  Quantum computing and entanglement for mathematicians. Representation theory and complex analysis, 345–376, Lecture Notes in Math., 1931, Springer, Berlin, 2008.

131. Adriano Garsia,Gregg Musiker, Nolan Wallach, Guoce Xin, Invariants, Kronecker products, and combinatorics of some remarkable Diophantine systems. Adv. in Appl. Math. 42 (2009), no. 3, 392–421.

132. Roe Goodman, Nolan R. Wallach,  Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.

133. Nolan Wallach, Oded Yacobi,  A multiplicity formula for tensor products of ${\rm SL}_2$SL2 modules and an explicit ${\rm Sp}_{2n}$Sp2n to ${\rm Sp}_{2n-2}\times{\rm Sp}_2$Sp2n-2×Sp2 branching formula. Symmetry in mathematics and physics, 151–155, Contemp. Math., 490, Amer. Math. Soc., Providence, RI, 2009.

134. Hanspeter Kraft,  Nolan R. Wallach, Polarizations and nullcone of representations of reductive groups. Symmetry and spaces, 153–167, Progr. Math., 278, Birkhäuser Boston, Inc., Boston, MA, 2010.

135. Gilad  Gour, Nolan R. Wallach, All maximally entangled four-qubit states. J. Math. Phys. 51 (2010), no. 11.

136. G. Gour and N. R. Wallach, Necessary and sufficient conditions for local manipulation of multipartite pure quantum states, New Journal of Physics 13(2011): 073013 (28 pp.).

Distributed Preprints:

1. N. R. Wallach:  “An elementary approach to theorems of  Paley-Wiener type on semi-simple Lie groups of split rank 1,” distributed preprint.

2. N. R. Wallach and T. J. Enright:  “The fundamental series of semi-simple Lie algebras and Lie groups,” distributed preprint.