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On the index of pseudo-differential operators on compact Lie groups. (English) Zbl 1421.35400

The author gives a formula for the index of an elliptic pseudo-differential operator \(A\) on a compact Lie group \(G\). With respect to the original work of M. F. Atiyah and I. M. Singer [Ann. Math. (2) 87, 484–530 (1968; Zbl 0164.24001)] for classical pseudo-differential operators on compact manifolds, here the author refers to [M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries. Background analysis and advanced topics. Basel: Birkhäuser (2010; Zbl 1193.35261)], where the operator \(A\) is expressed in terms of the Lie group structure. Consequently, the formula is based on a vector-valued symbol \(\sigma(A)\), mapping \(G\) into \(B(C^\infty(G))\). The author applies the so-called McKean-Singer lemma \[\text{ind}(A)= \text{tr}(e^{-tA^*A})- \text{tr}(e^{-tAA^*}),\quad t>0,\] which in the present case implies a local index formula of the form ind\((A)= \int_G\mu(g)\,dg\) for some density \(\mu\) on \(G\). As examples, the author provides a computation of the index on the torus, and on the group SU\((2)\) and SU\((3)\).

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
19K56 Index theory
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
58J20 Index theory and related fixed-point theorems on manifolds
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References:

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