C# how to detect generic type dynamically

I have a method in which I like to perform a few different casts using reflection mechanism.

private static T GetObject<T>(Dictionary<string, object> dict)
{
    Type type = typeof(T);
    var obj = Activator.CreateInstance(type);
    foreach (var kv in dict)
    {
        var p = type.GetProperty(kv.Key);
        Type t = p.PropertyType;
        Type t2 = kv.Value.GetType();
        if (t == t2)
        {
            p.SetValue(obj, kv.Value);
        }
        else if (p.PropertyType == typeof(object))
        {
            p.SetValue(obj, GetObject<type of p>((Dictionary<string, object>) kv.Value)); //???
        }
        else
        {
            p.SetValue(obj, (type of p)(kv.Value)); //???                    
        }
    }
    return (T)obj;
}

The second cast could be solved by iterating over primitives using an IF statement, but for the first cast it is a problem because a type can have a lot of properties of different types.

How can I detect dynamically the necessary cast/casts?