01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
9.06.1960
E-mail:
Keywords:
lie groups,
Lie algebras,
representation theory of Lie groups and Lie algebras,
intermediate between classical Lie groups and Lie algebras and their representations,
Verma modules,
highest weight modules,
analogues of H. Weyl formulae for characters and dimensions of intermediate Lie groups,
intertwining operator,
simple spectrum representation,
hyperbolic harmonics,
continuous basis,
the generalization of Funk-Hecke theorem.
Subject:
Area of my research is Lie groups, Lie algebras and Representation Theory. The basic content of my Ph.D. thesis (1987) consists in description of finite-dimensional representations of Lie group $Sp(2n-1)$ and in separation of multiple points of the spectrum in reduction $Sp(2n)$ to $Sp(2n-2)$. The category of (reducible, generally speaking) $Sp(2n-1)$-modules constructed there is similar to the category of irreducible modules for classical Lie groups. I proved in particular formulae for characters and dimensions of such $Sp(2n-1)$-modules that are analogues of H. Weyl well-known formulae. Modules of this category are cyclic and are found in 1&ndasth;1 correspondence with the set of intermediate rows in D. P. Zhelobenko branching rule for reduction $Sp(2n)\downarrow Sp(2n-2)$ (1962). Afterwards I generalized my construction to other series of intermediate (between classical) Lie groups: in 1994 to $A_{n-1/2}$, in 1998 to $B_{n-1/2}$, in 2001 to $D_{n-1/2}$.
Biography
From 1977 to 1982 — study at the Faculty of Mathematics and Mechanics of Moscow State University. From 1982 to 1985 — study at post-graduate courses by the Faculty of Mathematics and Mechanics of Moscow State University. Ph.D. degree in 1987. Ph.D. thesis "Spectral analysis of finite-dimensional representations of symplectic Lie groups and Lie algebras". Scientific adviser — prof. A. A. Kirillov. From 1986 I work as a lecturer of mathematics at the Donetsk National University. From 1994 to the present time I work as Associate Professor at that same University.
Main publications:
Shtepin V. V. Separation of multiple points of the spectrum in reduction $sp(2n)\downarrow sp(2n-2)$ // Funct. Anal. Appl., 1986, 20, 336–338.
Shtepin V. V. On a class of finite-dimensional $sp(2n-1)$-modules // Russian Math. Surveys, 1986, 41:3, 233–234.
Shtepin V. V. Intermediate Lie algebras and their finite-dimensional representations // Russ. Acad. Sci., Izv., Math. 43, 1994, no. 3, 559–579; translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, 1993, no.6, 176–198.
Shtepin V. V. The intermediate orthogonal Lie algebra ${\germ b}_{n-1/2}$ and its finite-dimensional representations // Russ. Acad. Sci., Izv. Math. 62, 1998, no. 3, 627–648; translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, 1998, no. 3, 201–223.
Shtepin V. V. Intermediate orthogonal Lie groups and their finite-dimensional representations. Proceedings of Ukrainian Congress of Mathematics. Kiev, 2001 (russian).
V. V. Shtepin, D. L. Konashenkov, “Characters and dimensions of highest-weight representations of the intermediate Lie group $D_{n-1/2}$”, Izv. RAN. Ser. Mat., 78:3 (2014), 205–224; Izv. Math., 78:3 (2014), 621–639
V. V. Shtepin, T. V. Shtepina, “An application of intertwining operators in functional analysis”, Izv. RAN. Ser. Mat., 73:6 (2009), 195–220; Izv. Math., 73:6 (2009), 1265–1288
V. V. Shtepin, “The intermediate Lie algebra $\mathfrak d_{n-1/2}$, the weight scheme and finite-dimensional representations with highest weight”, Izv. RAN. Ser. Mat., 68:2 (2004), 159–190; Izv. Math., 68:2 (2004), 375–404
V. V. Shtepin, “Separation of multiple points of spectrum in the reduction $\mathrm{sp}(2n)\downarrow\mathrm{sp}(2n-2)$”, Funktsional. Anal. i Prilozhen., 20:4 (1986), 93–95; Funct. Anal. Appl., 20:4 (1986), 336–338
V. V. Shtepin, “On a class of finite-dimensional $\operatorname{sp}(2n-1)$-modules”, Uspekhi Mat. Nauk, 41:3(249) (1986), 207–208; Russian Math. Surveys, 41:3 (1986), 233–234
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