Professor Shrawan Kumar

You may download a PDF version of my curriculum vitae here.

Here is a brief summary of my works in PDF format.

Contact Information:

Department of Mathematics
University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA
Telephone: 919-962-9615, Telefax: 919-962-2568
e-mail: shrawan [at]


Born in Ghazipur, U.P., INDIA.

Academic Qualifications:


Books Authored:


  1. A remark on universal connections

    Math. Ann. 260, 453-462 (1982). [pdf]

  2. A G-minimal model for principal G-bundles

    Annales de L'Institut Fourier, Grenoble 32, 205-219 (1982). [pdf]

  3. Geometry of Schubert cells and cohomology of Kac-Moody Lie algebras

    J. Diff. Geometry 20, 389-431 (1984). [pdf]

  4. Rational homotopy theory of flag varieties associated to Kac-Moody groups

    In: "Infinite dimensional groups with applications" (ed. by V. Kac). MSRI (Springer-Verlag) publications Vol. 4, 233-273 (1985). [pdf]

  5. Cohomology algebra of Schubert varieties associated to Kac-Moody groups

    In: "Proceedings of the 1984 Vancouver Conference in Algebraic Geometry" (ed. by J. Carrell et. al.). CMS Conference Proceedings Vol. 6, 277-299 (1986). [pdf]

  6. A generalization of the Conner conjecture and topology of Stein spaces dominated by C^n

    Topology 25, 483-493 (1986). [pdf]

  7. The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (with B. Kostant)

    Proc. Natl. Acad. Sci. USA 83, 1543-1545 (1986). [pdf]

  8. A homology vanishing theorem for Kac-Moody algebras with coefficients in the category O

    J. Algebra 102, 444-462 (1986). [pdf]

  9. Non-representability of cohomology classes by bi-invariant forms (Gauge and Kac-Moody groups)

    Commu. Math. Physics 106, 177-181 (1986). [pdf]

  10. The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (with B. Kostant)

    Advances in Math. 62, 187-237 (1986). [pdf]

  11. T-equivariant K-theory of generalized flag varieties (with B. Kostant)

    Proc. Natl. Acad. Sci. USA 84, 4351-4354 (1987). [pdf]

  12. Extension of the category O^g and a vanishing theorem for the Ext functor for Kac-Moody algebras

    J. Algebra 108, 472-491 (1987). [pdf]

  13. Demazure character formula in arbitrary Kac-Moody setting

    Invent. Math. 89, 395-423 (1987). [pdf]

  14. Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture

    Invent. Math. 93, 117-130 (1988). [pdf]

  15. Existence of certain components in the tensor product of two integrable highest weight modules for Kac-Moody algebras

    In: "Infinite dimensional Lie algebras and groups" (ed. by V.G. Kac). Advanced series in Mathematical Physics Vol. 7, World Scientific, 25-38 (1989). [pdf]

  16. A refinement of the PRV conjecture

    Invent. Math. 97, 305-311 (1989). [pdf]

  17. Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras

    Math. Annalen 286, 709-729 (1990). [pdf]

  18. A new geometric invariant associated to the space of flat connections (with K. Guruprasad)

    Compositio Math. 73, 199-222 (1990). [pdf]

  19. T-equivariant K-theory of generalized flag varieties (with B. Kostant)

    J. Diff. Geometry 32, 549-603 (1990). [pdf]

  20. Proof of Wahl's conjecture on surjectivity of the Gaussian map for flag varieties

    In: " Algebraic groups" (ed. by S. Ramanan). Proceedings of the Hyderabad conference. Manoj Prakashan, 275-278 (1991). [pdf]

  21. A geometric realization of minimal rak k-type of Harish-Chandra modules for complex s.s. groups (with B. Kostant)

    In: "Kazhdan-Lusztig theory and related topics" (ed. by V. Deodhar). Contemporary Mathematics vol. 139 published by AMS, 249-264 (1992). [pdf]

  22. Proof of Wahl's conjecture on surjectivity of the Gaussian map for flag varieties

    American J. of Math. 114, 1201-1220 (1992). [pdf]

  23. Cohomology of quantum groups at roots of unity (with V. Ginzburg)

    Duke Math. J. 69, 179-198 (1993). [pdf]

  24. Finiteness of local fundamental groups for quotients of affine varieties under reductive groups

    Comment. Math. Helvetici 68, 209-215 (1993). [pdf]

  25. Equivariant cohomology with generalized coefficients (with M. Vergne)

    Ast\'erisque 215, 109-204 (1993). [pdf]

  26. Toward proof of Lusztig's conjecture concerning negative level representations of affine Lie algebras

    J. Algebra 164, 515-527 (1994). [pdf]

  27. Symmetric and exterior powers of homogeneous vector bundles

    Math. Annalen 299, 293-298 (1994). [pdf]

  28. Representations of quantum groups at roots of unity

    In: "Quantum topology" (ed. by D.N. Yetter). World Scientific, Singapore, 187-224 (1994). [pdf]

  29. Infinite Grassmannians and moduli spaces of G-bundles (with M. S. Narasimhan and A. Ramanathan)

    Math. Annalen 300, 41-75 (1994). [pdf]

  30. The nil Hecke ring and singularity of Schubert varieties

    In: "Lie theory and geometry (in honor of B. Kostant)" (ed. by J.-L. Brylinski et. al.). Progress in Math. vol. 123, Birkhauser, 497-507 (1994). [pdf]

  31. The nil Hecke ring and singularity of Schubert varieties

    Inventiones Math. 123, 471-506 (1996). [pdf]

  32. Shapovalov determinant for restricted and quantized restricted enveloping algebras (with G. Letzter)

    Pacific J. of Math. 179, 123-161 (1997). [pdf]

  33. Infinite Grassmannians and moduli spaces of G-bundles

    In: "Vector Bundles on Curves- New Directions," (ed. by M.S. Narasimhan). Springer Lecture Notes in Math. vol. 1649, 1-49 (1997). [pdf]

  34. Picard group of the moduli spaces of G-bundles (with M.S. Narasimhan)

    Math. Annalen 308, 155-173 (1997). [pdf]

  35. Fusion product of positive level representations and Lie algebra homology

    In: "Geometry and physics" (ed. by J.E. Andersen et. al.). Lecture Notes in Pure and Applied Mathematics vol. 184, Marcel Dekker, Inc., 253-259 (1997). [pdf]

  36. Positivity of the cup product in cohomology of flag varieties associated to Kac-Moody groups (with M.V. Nori)

    Inter. Math. Res. Not. 14, 757-763 (1998). [pdf]

  37. Frobenius splitting of cotangent bundles of flag varieties (with N. Lauritzen and J. Thomsen)

    Inventiones Math. 136, 603--621 (1999). [pdf]

  38. Homology of certain truncated Lie algebras

    In: "Recent Developments in Quantum Affine Algebras and Related Topics," (ed. by K.C. Misra et. al.). Contemporary Mathematics vol. 248, 309--325 (1999), published by AMS. [pdf]

  39. Frobenius splitting in characteristic zero and the quantum Frobenius map (with P. Littelmann)

    J. of Pure and applied Algebra 152, 201--216 (2000). [pdf]

  40. Frobenius splitting of Hilbert schemes of points on surfaces (with J.F. Thomsen)

    Math. Annalen 319, 797--808 (2001). [pdf]

  41. Algebraization of Frobenius splitting via quantum groups (with P. Littelmann)

    Annals of Mathematics 155, 491--551 (2002). [pdf]

  42. A conjectural generalization of the n! result to arbitrary groups (with J.F. Thomsen)

    Transformation Groups 8, 69--94 (2003). [pdf]

  43. Equivariant analogue of Grothendieck's theorem for vector bundles on \Bbb P^1

    In: "Perspectives in Geometry and Representation Theory (A tribute to C.S. Seshadri)," Hindustan Book Agency, 500--501 (2003). [pdf]

  44. The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type (with B. Leeb and J. Millson)

    Contemporary Mathematics vol. 332, published by AMS, 171--195 (2003). [pdf]

  45. A new realization of the cohomology of Springer fibers (with J.F. Thomsen)

    In: "Proceedings of the International Conference on Algebraic Groups and Arithmetic" (Volume dedicated to M.S. Raghunathan on his sixtieth birthday), Narosa Publishing House, 119--125 (2004). [pdf]

  46. Induction functor in noncommutative equivariant cohomology and Dirac cohomology

    J. Algebra 291, 187--207 (2005). [pdf]

  47. Explicit determination of the Picard group of moduli spaces of semi-stable G-bundles on curves (with A. Boysal)

    Math. Annalen 332, 823--842 (2005). [pdf]

  48. Extensions of algebraic groups (with K-H. Neeb)

    In:"Studies in Lie Theory" (Volume dedicated to A. Joseph on his sixtieth birthday), Progress in Mathematics vol. 243, Birkhauser, 365--376 (2006). [pdf]

  49. Eigenvalue problem and a new product in cohomology of flag varieties (with P. Belkale)

    Inventiones Math. 166, 185--228 (2006). [pdf]

  50. On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

    J. of Am. Math. Soc. 21, 797--808 (2008). [pdf]

  51. The eigencone and saturation for Spin(8) (with M. Kapovich and J. Millson)

    Pure and Applied Math. Quarterly 5, 755--780 (2009). [pdf]

  52. Descent of line bundles to GIT quotients of flag varieties by maximal torus

    Transformation Groups 13, 757--771 (2008). [pdf]

  53. Special isogenies and tensor product multiplicities (with J. Stembridge)

    International Mathematics Research Notices, vol. 2007 (no. 20), 1--13 (2007). [pdf]

  54. Eigencone, saturation and Horn problems for symplectic and odd othogonal groups (with P. Belkale)

    J. Alg. Geom. 19, 199--242 (2010). [pdf]

  55. Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups (with G. Lusztig and D. Prasad)

    Contemporary Math. vol. 478, 99--101 (2009). [pdf]

  56. On positivity in T-equivariant K-theory of flag varieties (with W. Graham)

    International Mathematical Research Notices, vol. 2008, 1--43 (August, 2008). [pdf]

  57. A conjectural presentation of fusion algebras (with A. Boysal)

    In: "Algebraic Analysis and Around" (in honor of M. Kashiwara's sixtieth birthday). Advanced Studies in Pure Mathematics vol. 54, 95--107 (2009). [pdf]

  58. A generalization of Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces

    Math. Annalen 346, 67--84 (2010). [pdf]

  59. Finiteness of the number of compatibly split subvarieties (with V. Mehta)

    International Mathematics Research Notices, vol. 2009 (no. 19), 3595--3597 (2009). [pdf]

  60. Tensor Product Decomposition

    In: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010. [pdf]

  61. A generalization of Fulton's conjecture for arbitrary groups (with P. Belkale and N. Ressayre)

    Math. Annalen 354, 401-425 (2012). [pdf]

  62. Geometry of orbits of permanents and determinants

    Comment. Math. Helv. 88, 759--788 (2013). [pdf]

  63. An algebro-geometric realization of equivariant cohomology of some Springer fibers (with C. Procesi)

    J. Algebra 368, 70--74 (2012). [pdf]

  64. An approach towards the Kollar-Peskine problem via the Instanton Moduli Space

    Proceedings of Symposia in Pure Mathematics 86, 217-225 (2012). [pdf]

  65. Richardson varieties have Kawamata log terminal singularities (with K. Schwede)

    International Mathematics Research Notices, rns241, (2012), 23 pages [pdf]

  66. Positivity in T-Equivariant K-Theory of flag varieties associated to Kac-Moody groups

    J. Eur. Math. Soc. 19, 2469--2519 (2017). [pdf]

  67. A study of the representations supported by the orbit closure of the determinant

    Compositio Math 151, 292-312 (2015). [pdf]

  68. Dimension of zero weight space: an algebro-geometric approach (with D. Prasad)

    J. Algebra 403, 324--344 (2014). [pdf]

  69. A study of saturated tensor cone for symmetrizable Kac-Moody algebras (with M. Brown)

    Math. Annalen 360, 901-936 (2014). [pdf]

  70. A survey of the additive eigenvalue problem (with an appendix by M. Kapovich)

    Transformation Groups 19, 1051-1148 (2014). [pdf]

  71. Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra (with N. Bushek)

    Differential Geometry and its Applications 35, 210--223 (2014). [pdf]

  72. The multiplicative eigenvalue problem and deformed quantum cohomology (with P. Belkale)

    Advances in Mathematics 288, 1309--1359 (2016). [pdf]

  73. Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group (with J.M. Landsberg)

    Discrete Mathematics 338, 1232--1238 (2015). [pdf]

  74. Additive Eigenvalue Problem

    EMS Newsletter (2015), 8 pages. [pdf]

  75. Components of V(ρ) ⊗ V(ρ) (with R. Chirivi and A. Maffei)

    Transformation Groups, 22, 645--650 (2017). [pdf]

  76. Representation ring of Levi subgroups versus cohomology ring of flag varieties

    Math. Annalen 366, 395--415 (2016). [pdf]

  77. Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups II (with S. Baldwin)

    Representation Theory (An electronic journal of the American Mathematical Society) 21, 35--60 (2017). [pdf]

  78. A complete set of intertwiners for arbitrary tensor product representations via current algebras

    Transformation Groups 24, 115--125 (2019). [pdf]

  79. Conformal blocks for Galois covers of algebraic curves (with Jiuzu Hong)

    Compositio Math. 159, 2191--2259 (2023) [pdf]

  80. On the faces of the tensor cone of symmetrizable Kac-Moody Lie algebras (with Nicolas Ressayre), Preprint (2024), 45 pages.


  81. Representation ring of Levi subgroups versus cohomology ring of flag varieties II

    J. Algebra 556, 340--362 (2020). [pdf]

  82. Elliptic classes of Schubert varieties (with Richard Rimanyi and Andrzej Weber)

    Math. Annalen 378, 703--728 (2020). [pdf]

  83. Root components for tensor product of affine Kac-Moody Lie algebra modules (with Samuel Jeralds)

    Representation Theory (An electronic journal of the American Mathematical Society) 26, 825--858 (2022). [pdf]

  84. Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras, Journal of Algebra 594, 307--312 (2022).


  85. Category C_k of multi-loop algebra representations versus modular representations: Questions of G. Lusztig, Preprint (2022), 14 pages.


  86. Representation ring of Levi subgroups versus cohomology ring of flag varieties III (with Jiale Xie), Journal of Pure and Applied Algebra 228, (2024).


  87. Twisted conformal blocks and their dimension (with Jiuzu Hong), Math. Zeitschrift, 29 pages (2024).


  88. Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells, International Mathematics Research Notices (2023), 12 pages.


  89. Components of V(ρ) ⊗ V(ρ) and dominant weight polyhedra for affine Kac-Moody Lie algebras (with Sam Jeralds), Transformation Groups (2023), 22 pages.


  90. Lie algebra cohomology of the positive part of twisted affine Lie algebras (with Jiuzu Hong), Preprint (2023), 30 pages.


  91. Positivity in T-Equivariant K-theory of partial flag varieties associated to Kac-Moody groups (with Joseph Compton), Preprint (2023), 25 pages.


  92. Nonexistence of regular maps between homogeneous projective varieties, Preprint (2023), 6 pages.


  93. Errata for the book ‘Conformal Blocks, Generalized Theta Functions and the Verlinde Formula’ by Shrawan Kumar, 5 pages (2024).


  94. Erratum to the paper `Conformal blocks for Galois covers of algebraic curves' published in Compositio Mathematica, volume 159 (2023), 2291--2259 (with Jiuzu Hong), 2 pages (2024).


Lecture Notes for Some Talks:

  1. Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

    (Talk given at AMS Sectional meeting held at Davidson College, March 2007) [pdf]

  2. Geometry of Schubert varieties and Demazure character formula

    (Talks given at the Hausdorff Research Institute for Mathematics, Bonn, Germany, April 2011) [pdf]

  3. Borel-Weil-Bott theorem and geometry of Schubert varieties

    (Lectures given at the MRC Meeting, Snowbird, Utah, June 2012) [pdf]

  4. Equivariant K-theory of flag varieties

    (Talks given at Jacobs University, Bremen (Germany), August 2012) [pdf]