  In mid 60s Bott proved that (1) the index theorem for homogeneous,
G-invariant, elliptic differential operators acting in the spaces of sections
of induced representations of G over G/H reduces to the Weyl character formula
and (2) the index of an equivariant elliptic operator does not depend on the
operator, but on the representations. Here the same theorem is formulated for
the unitary supergroup G=U(p|q). For atypical representations the character
formula does not reduce to that for the Lie group underlying the supergroup G
and this contradicts a statement of Rempel and Schmitt on index on
supermanifolds (Pseudodifferential operators and the index theorem on
supermanifolds. Seminar Analysis, 1981/82, 92--131, Akad. Wiss. DDR, Berlin,
1982; id., Pseudodifferential operators and the index theorem on
supermanifolds. Math. Nachr. 111 (1983), 153--175).
