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Let F be a local field, nonarchimedean and of characteristic not 2. Let (V,Q) be a nondegenerate quadratic space over F, of dimension n. Let Mr be the direct sum of r copies of V . We prove that, for r < n there is no nonzero distribution on Mr which under the action of the orthogonal group transforms according to the character determinant. Let F be a local field, nonarchimedean and of characteristic not 2. Let V be a vector space over F of finite dimension n ≥ 1 equipped with a nondegenerate quadratic form Q. Let G = O(Q) be the orthogonal group of Q and χ the character of G such that χ(g) = 1(resp. χ(g) = −1) if the determinant of g is 1 (resp. −1). Let r ≥ 1 be an integer and let Mr be the direct sum of r copies of V or equivalently the set of linear maps from F onto V or, choosing a basis of V , the space of n × r matrices with coefficients in F. We let G act on Mr on the left and GLr(F) act on the right. Finally, let S(Mr) be the space of locally constant functions on Mr with compact support and values in C and its dual S (Mr) the space of distributions on Mr. It will be convenient to include the case r = 0 with M0 = (0) and the trivial action of G. Theorem. Suppose that r < n. If T ∈ S (Mr) transforms according to the character χ under the action of G = O(Q), then T = 0. This result was stated in Appendix 2 in [R] and used in the proof of the theorem in Chapter II. However, the proof given there was confusing to the reader due to incomplete detail. As the result itself is interesting we present here, in this short note, a full version of our proof. Note that this is false if r ≥ n. Indeed, suppose first that r = n and let x = (ξ1, . . . , ξn) be an orthogonal basis of V . Then Q(ξi) = αi = 0 (the quadratic form is nondegenerate) and by Witt Theorem the orbit of x under G is the set of all orthogonal systems (ηi, . . . , ηn) such that for all i, Q(ηi) = αi. In particular, it is a closed orbit. Now the isotropy subgroup of x in G is trivial so that the orbit is homeomorphic to G. Choose a Haar measure dg on G. The distribution χ(g)dg is a nontrivial distribution on G transforming according to the character χ. We may view this distribution as a distribution on Gx, hence on V . If r > n we note that Mn is imbedded into Mr by (ξ1, . . . , ξn) → (ξ1, . . . , ξn, 0, . . . , 0) as a closed invariant subset and, therefore, S (Mn) ⊆ S (Mr). Received by the editors November 8, 2011 and, in revised form, August 20, 2012 and October 2, 2012. 2010 Mathematics Subject Classification. Primary 22E55. Sadly, the author passed away on April 17, 2012. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication

Complement to the appendix of: “On the Howe duality conjecture”