mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the affine Grassmannian of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''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 the ...

—a colimit of finite-dimensional schemes—which can be thought of as a flag variety 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 fi ...

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

k\text

and

\mathrm

the category of commutative ''k''-algebras and the category of sets respectively. Through the

Yoneda lemma In mathematics, the Yoneda lemma is a fundamental 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 (viewing a group as a ...

, a scheme ''X'' over a field ''k'' is determined by its functor of points, which is the functor

X:k\text \to \mathrm

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 Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...

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, 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 cu ...

''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 that has a finite kernel and is a ...

, Gr_''G'' is in fact ind-projective, i.e., an inductive limit of projective schemes.

Definition as a coset space

Let us denote by

\mathcal K = k((t))

the field of formal Laurent series over ''k'', and by

\mathcal O = k t

the ring of formal

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

over ''k''. By choosing a trivialization of ''E'' over all of

\operatorname\mathcal O

, the set of ''k''-points of Gr_''G'' is identified with the coset space

G(\mathcal K) / G(\mathcal O)

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