#### Glasnik Matematicki, Vol. 34, No.2 (1999), 147-185.

### PARABOLIC INDUCTION AND JACQUET MODULES
OF REPRESENTATIONS OF *O*(2*n*, *F*)

### Dubravka Ban

Department of Mathematics, Purdue University, West Lafayette,
IN 47907, USA

*e-mail:* `dban@mapmf.pmfst.hr`;
`dban@math.purdue.edu`

**Abstract.** For the sum of the Grothendieck groups
of the categories of smooth finite length representations of
*O*(2*n*, *F*) (resp., *SO*(2*n*, *F*)),
*n* ≥ 0,
(*F* a p-adic field), the structure of a module and a
comodule over the sum of the Grothendieck groups of the categories
of smooth finite length representations of
*GL*(*n*, *F*), *n* ≥ 0,
is achieved.
The multiplication is defined in terms of parabolic induction,
and the comultiplication in terms of Jacquet modules. Also, for even
orthogonal groups, the combinatorial formula, which connects the module
and comodule structures, is obtained.

**1991 Mathematics Subject Classification.**
20G05, 22E50.

**Key words and phrases.** Representations of p-adic groups,
even orthogonal groups, special even orthogonal groups, parabolic
induction, Jacquet modules.

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