In
mathematics, the affine Grassmannian of an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
''G'' over a field ''k'' is an
ind-scheme In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.
Examples
*\mathbbP^ = \varinjlim \mathbbP^N is an ind-scheme.
*Perhaps th ...
—a colimit of finite-dimensional
schemes—which can be thought of as a
flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
for the loop group ''G''(''k''((''t''))) and which describes the representation theory of the
Langlands dual
In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a fie ...
group
''L''''G'' through what is known as the
geometric Satake correspondence.
Definition of Gr via functor of points
Let ''k'' be a field, and denote by
and
the category of commutative ''k''-algebras and the category of sets respectively. Through the
Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...
, a scheme ''X'' over a field ''k'' is determined by its
functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...
, which is the functor
which takes ''A'' to the set ''X''(''A'') of ''A''-points of ''X''. We then say that this functor is
representable by the scheme ''X''. The affine Grassmannian is a functor from ''k''-algebras to sets which is not itself representable, but which has a
filtration by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it.
Let ''G'' be an algebraic group over ''k''. The affine Grassmannian Gr
''G'' is the functor that associates to a ''k''-algebra ''A'' the set of isomorphism classes of pairs (''E'', ''φ''), where ''E'' is a
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
for ''G'' over Spec ''A'' and ''φ'' is an isomorphism, defined over Spec ''A''((''t'')), of ''E'' with the trivial ''G''-bundle ''G'' × Spec ''A''((''t'')). By the
Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by .
The theorem
...
, it is also possible to specify this data by fixing an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
''X'' over ''k'', a ''k''-point ''x'' on ''X'', and taking ''E'' to be a ''G''-bundle on ''X''
''A'' and ''φ'' a trivialization on (''X'' − ''x'')
''A''. When ''G'' is a
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
, Gr
''G'' is in fact ind-projective, i.e., an inductive limit of projective schemes.
Definition as a coset space
Let us denote by
the field of
formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
over ''k'', and by
the ring of formal power series over ''k''. By choosing a trivialization of ''E'' over all of
, the set of ''k''-points of Gr
''G'' is identified with the coset space
.
References
*{{cite book, author=Alexander Schmitt, title=Affine Flag Manifolds and Principal Bundles, url=https://books.google.com/books?id=xrPoBAdiVdQC&pg=PA1, accessdate=1 November 2012, date=11 August 2010, publisher=Springer, isbn=978-3-0346-0287-7, pages=3–6
Algebraic geometry