View source for Transversal elliptic operator

''transversally elliptic operator''

A differential or [[Pseudo-differential operator|pseudo-differential operator]] (cf. also [[Differential operator|Differential operator]]) commuting with the action of some Lie group on a manifold on which the operator is defined and which is elliptic in the directions normal to the orbits of this group. If the operator acts on sections of vector bundles, then it is also assumed that the action of the given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939701.png" /> is lifted to each of the bundles and, further, is extended to sections of the bundles. If the group is discrete, then a transversally elliptic operator is an ordinary elliptic operator commuting with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939702.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939703.png" /> acts transitively on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939704.png" />, then any differential or pseudo-differential operator commuting with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939705.png" /> is a transversally elliptic operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939707.png" /> are compact, then the index is defined, and can be calculated, for a transversally elliptic operator. It is a generalized function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939708.png" /> (see [[Index formulas|Index formulas]]). Refined spectral characteristics of a self-adjoint transversally elliptic operator, taking into account the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093970/t0939709.png" />, can be defined.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,   "Elliptic operators and compact groups" , ''Lect. notes in math.'' , '''401''' , Springer  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Brüning,   E. Heintze,   "Representations of compact Lie groups and elliptic operators"  ''Invent. Math.'' , '''50'''  (1979)  pp. 169–203</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Shubin,   "Spectral properties and distribution function of the spectrum of a transversally elliptic operator"  ''Trudy Sem. Petrovsk.'' , '''8'''  (1982)  pp. 239–258  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Smagin,   M.A. Shubin,   "The zeta function of a transversally elliptic operator"  ''Siber. Math. J.'' , '''25''' :  6  (1984)  pp. 959–966  ''Siber. Mat. Zh.'' , '''25''' :  6  (1984)  pp. 158–166</TD></TR></table>



====Comments====
So, a (pseudo-)differential operator is transversally elliptic if the combination of the operator with the vector fields defined by the infinitesimal action of the group form an overdetermined elliptic system (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]).