Continuous crystal and Duistermaat-Heckman measure for Coxeter groups

Advances in Mathematics

Volumn 221, Issue 5, 2009, Pages 1522-1583

  Biane, Philippe ^a   Bougerol, Philippe ^b   O'Connell, Neil ^c  

^a CNRS   (France)

^b SORBONNE UNIVERSITÉ   (France)

^c UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Brownian motion; Coxeter groups; Representation theory

Indexed keywords

EID: 67349106244     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2009.02.016     Document Type: Article

References (32)

1. Alexeev V., and Brion M. Toric degenerations of spherical varieties. Selecta Math. (N.S.) 10 4 (2004) 453-478
2. Berenstein A., and Zelevinsky A. Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143 1 (2001) 77-128
3. Biane Ph., Bougerol Ph., and O'Connell N. Littelmann paths and Brownian paths. Duke Math. J. 130 1 (2005) 127-167
4. Bourbaki N. Groupes et algèbres de Lie, Ch. IV-VI (1975), Hermann, Paris
5. Caldero P. Toric degenerations of Schubert varieties. Transform. Groups 7 1 (2002) 51-60
6. Dooley A.H., Repka J., and Wildberger N.J. Sums of adjoint orbits. Linear Multilinear Algebra 36 (1993) 79-101
7. Duistermaat J.J., and Heckman G.J. On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 2 (1982) 259-268
8. Fomin S., and Zelevinsky A. Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 2 (1999) 335-380
9. Fulton W. Young Tableaux. London Math. Soc. Stud. Texts vol. 35 (1997), Cambridge Univ. Press, Cambridge
10. Gross K., and Richards D. Total positivity, finite reflection groups, and a formula of Harish-Chandra. J. Approx. Theory 82 (1995) 60-87
11. Harish-Chandra. A formula for semisimple Lie groups. Amer. J. Math. 79 (1957) 733-760
12. Helgason S. Geometric Analysis on Symmetric Spaces. Math. Surveys Monogr. vol. 39 (1994), Amer. Math. Soc., Providence, RI
13. Henriques A., and Kamnitzer J. The octahedron recurrence and gl (n) crystals. Adv. Math. 206 1 (2006) 211-249
14. Henriques A., and Kamnitzer J. Crystals and coboundary categories. Duke Math. J. 132 (2006) 191-216
15. Hong J., and Kang S.-J. Introduction to Quantum Groups and Crystal Bases. Grad. Stud. Math. vol. 42 (2002), Amer. Math. Soc., Providence, RI
16. Humphreys J.E. Reflection Groups and Coxeter Groups. Cambridge Stud. Adv. Math. vol. 29 (1990), Cambridge Univ. Press, Cambridge
17. Joseph A. Quantum Groups and Their Primitive Ideals (1995), Springer, Berlin
18. Joseph A. Lie algebras, their representations and crystals. http://www.wisdom.weizmann.ac.il/gorelik/agrt.htm
19. Joseph A. A Pentagonal Crystal, the Golden section, alcove packing and aperiodic tilings. arXiv:0811.0336
20. Kashiwara M. The crystal base and Littelmann's refined Demazure character formula. Duke Math. J. 71 3 (1993) 839-858
21. Kashiwara M. Bases cristallines des groupes quantiques. rédigé par Charles Cochet. Cours Spécialisés SMF vol. 9 (2002), Soc. Math. France, Paris 115 pp
22. n. Lett. Math. Phys. 35 4 (1995) 374-395
23. Littelmann P. Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995) 499-525
24. Littelmann P. Cones, crystals, and patterns. Transform. Groups 3 2 (1998) 145-179
25. Lusztig G. Introduction to Quantum Groups (1994), Birkhäuser, Basel
26. Moody R.V., and Patera J. Quasicrystals and icosians. J. Phys. A 26 (1993) 2829-2853
27. Morier-Genoud S. Geometric lifting of the canonical basis and semitoric degenerations of the Richardson varieties. Trans. Amer. Math. Soc. 360 (2008) 215-235
28. O'Connell N., and Yor M. Brownian analogues of Burke's theorem. Stochastic Process. Appl. 96 (2001) 285-304
29. Opdam E.M. Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compos. Math. 85 (1993) 333-373
30. Pitman J.W. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 (1975) 511-526
31. Rösler M. Positivity of Dunkl intertwining operator. Duke Math. J. 98 (1999) 445-463
32. Rösler M. A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355 (2003) 2413-2438