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MATHEMATICAL TRIPOS Part III
Thursday 31 May 2007 9.00 to 12.
PAPER 1
SMOOTH REPRESENTATION
THEORY OF P -ADIC GROUPS
Attempt THREE questions. There are FOUR questions in total.
The questions carry equal weight.
In the following questions, F always denotes a non-Archimedean local field with ring of integers o and maximal ideal p = $o. The valuation | · | on F is normalized by |$| = q−^1 , where q is the cardinality of the residue field of F. The notation diag(a 1 ,... , an) denotes an n-by-n square matrix (aij ) all of whose non-diagonal entries aij are zero, and aii = ai, i = 1,... , n.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None Treasury Tag Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1 (a) Let ι : F → C be an embedding of fields and n > 1 a natural number. Is the induced homomorphism π : GLn(F ) → GLn(C), (aij ) 7 → (ι(aij )), a smooth representation? Explain your answer.
(b) Let G be an `-group, V = C c∞ (G, C) the space of locally constant functions with compact support on G. Show that the representation ρ : G → GL(V ), (ρ(g)f )(x) = f (xg), is smooth. Show furthermore that it is admissible if and only if G is compact.
2 For each natural number i > 1 let χi be a complex-valued character of F ×^ which is trivial on 1 + pi+1^ but not trivial on 1 + pi. Let (Vi, χi) be the one-dimensional representation of F ×^ on C given by the character χi.
(a) Let (V, π) be the representation of F ×^ which is the direct sum of the represen- tations (Vi, χi), i.e.
V =
i> 1
Vi.
Is V an admissible representation of F ×? Explain your answer.
(b) Show that the representation π∗^ on the algebraic dual space V ∗^ = HomC(V, C) is naturally isomorphic to (^) ∏
i> 1
V (^) i∗ ,
where V (^) i∗ is the one-dimensional representation given by the character χ− i 1.
(c) Show that the smooth dual V ∨^ of V , i.e. the subrepresentation of V ∗^ consisting of all smooth vectors, is
V ∨^ =
i> 1
V (^) i∗.
3 Let n > 2 be a natural number, B ⊂ GLn(F ) be the subgroup of upper-triangular matrices, U ⊂ B the normal subgroup of upper-triangular matrices having 1’s on the diagonal, and T ⊂ B the subgroup of diagonal matrices. Let (V, ρ) be an admissible representation of B.
(a) Let δ = diag($−(n−1), $−(n−2),... , 1), and K ⊂ B be a compact-open subgroup of the form T 0 U 0 with compact-open subgroups T 0 ⊂ T and U 0 ⊂ U. For i > 0 put Ki = δiKδ−i. Show that the map
V K^ → V Ki^ , v 7 → ρ(δi)(v) ,
is an isomorphism. Show further that K is contained in Ki for i 0. Using this and a dimension argument, deduce that V K^ = V Ki^ for all sufficiently large i.
(b) Use (a) to prove that U acts trivially on V.
Paper 1
