(Apartments and affine Weyl groups) Bourbaki, Lie groups and Lie algebras,
Chapters 4-6.
P. Garrett, Buildings and classical groups, Chapman and Hall, 1997.
K. Brown, Buildings, Springer-Verlag, 1989.
Information about study topics (last updated March 20)
Material covered Jan 10:
Recall definitions of differentiable manifold, tangent space,
tangent bundle, smooth maps, differentials.
Definition of Lie group; Examples:
compact one-dimensional torus, general linear groups;
Definition of Lie algebra, identification of Lie algebra
with the tangent space at the identity.
Material covered on Jan 15 (apart from the example, material is
from Warner):
Use of Lie bracket of smooth vector fields to define bracket
for Lie algebra of Lie group.
Definitions of Lie group homomorphism and Lie subgroup.
Using differential of Lie group homomorphism to define
a homomorphism of Lie algebras.
Lie algebra of n by n general linear group.
Example of homomorphism SU(2) onto SO(3) whose differential
is an isomorphism of Lie algebras.
Material covered on Jan 17:
Existence of connected
Lie sugroups corresponding
to Lie subalgebras (proof uses theorems about integral manifolds
of distributions).
General comments about uniqueness of differentiable
structure for abstract subgroups of Lie groups that
happen to be submanifolds relative to inclusion.
Material covered on Jan 22:
Some comments about abstract subgroups of Lie groups
that are submanifolds relative to inclusion.
Theorem about a Lie subgroup having closed image
if and only if the submanifold immersion is an imbedding.
Material covered on Jan 24 and Jan 29:
General remarks about universal covering
groups.
Definition and basic properties of exponential map.
Identifying subgroups as Lie subgroups using exponential maps.
Matrix exponentials and some examples of closed Lie subgroups of
general linear groups.
Comments about algebraic groups - exponentials
of matrices with entries in the p-adic numbers may not converge.
Material covered on Jan 31:
Continuous homomorphisms of Lie groups are smooth.
Closed subgroups of Lie groups are Lie subgroups.
Kernels of Lie group homomorphisms are closed Lie subgroups.
Definition of action of a Lie group on a manifold.
The adjoint representation of a Lie group.
Material covered on Feb 5:
The differential of the adjoint representation of the Lie group
is the adjoint representation of its Lie algebra.
Normal subgroups in Lie groups and ideals in Lie algebras.
The centre of a Lie group is a closed subgroup.
If G is connected, the centre of G is the kernel of the adjoint
representation.
A connected Lie group is abelian if and only if its Lie algebra
is abelian.
Material covered on February 7:
General remarks about homogeneous manifolds.
Using homogeneous manifolds to demonstrate connectedness
of some Lie groups.
(Remark: This finishes the material from Warner.)
Example of a Lie group that is not a closed linear group.
Representations of compact groups - comments about general theory.
Material covered on February 12:
Remarks about Haar measure.
Remarks about representations of compact groups.
Using the Peter-Weyl Theorem to prove that a compact
Lie group is isomorphic to a closed linear group.
Material covered on February 14:
Compact Lie algebras; Lie algebras of compact groups
are reductive.
G compact connected Lie group: G is a product of a semisimple
closed subgroup
and the centre of G.
Maximal tori - definition and a few examples.
Material covered on February 26:
Conjugacy of maximal tori in compact connected Lie groups.
Comments about root systems associated to maximal
tori in compact Lie groups.
Material covered on February 28:
Theorem: T maximal torus in a compact connected Lie group G.
Every conjugacy class in G intersects T.
More properties of tori.
Weyl groups as normalizers of maximal tori.
Parametrization of conjugacy classes via the Weyl group.
Weyl groups of root systems associated to maximal tori.
Material covered on March 4:
Comparing the Weyl group of a maximal torus
and the Weyl group of the root system.
Material covered on March 6:
Classification of irreducible finite-dimensional
representations of compact Lie groups.
Analytically integral forms are algebraically integral.
Material covered on March 11:
Weyl's character formula (characters of representations of compact Lie groups).
Material covered on March 13:
Statement of Weyl's integration formula for compact Lie groups.
Outline of use of the integration formula
to prove
of the classification theorem and Weyl character formula for representations
of compact Lie groups.
Borel subalgebras, example of parabolic subalgebras in
4 by 4 symplectic Lie algebras.
Material covered on March 18:
Reductive groups (algebraic groups and Lie groups).
Algebraic tori in algebraic groups and Lie groups.
Parabolic subgroups of split groups.
Statement of Bruhat Decomposition.
Statement of Iwasawa decomposition (example: special linear groups
over local fields).
Material covered on March 20:
General remarks about integration formulas related to
parabolic subgroups.
Comments about the role of parabolic subgroups in representation theory.Example: SO(5,3) (quasi-split example) relative roots and parabolic
subgroups.
Material covered on March 25:
Existence of Cartan decompositions of semisimple Lie algebras.
Material covered on March 27:
Cartan decompositions for semisimple Lie groups.
Material covered on April 1:
Use of Cartan decomposition to produce relative root system.
Weyl group representatives realized by elements of Ad K.
Material covered on April 3:
Maximally split algebraic tori in semisimple algebraic groups.
Relative root systems.
Definition of parabolic subalgebras and subgroups of semisimple groups.
General comments about Satake diagrams, classification of semisimple groups.
Material covered on April 8:
Root data and general comments about L-groups.
Upcoming material (April 10):
Affine roots, affine Weyl groups, apartments, buildings,
and structure of reductive groups over p-adic fields.
Compact subgroups and points in buildings, parahoric subgroups.